ISOMETRY GROUPS AND LATTICES OF NON-POSITIVELY CURVED...

ISOMETRY GROUPS AND LATTICES

OF NON-POSITIVELY CURVED SPACES

PIERRE-EMMANUEL CAPRACE* AND NICOLAS MONOD‡

Abstract. We develop the structure theory of full isometry groups of locally compactnon-positively curved metric spaces and their discrete subgroups. Classical results onHadamard manifolds are extended to that broad geometric setting; properties that singleout semi-simple groups and their lattices are highlighted. Amongst the discussed themesare de Rham decompositions, normal subgroup structure, characterising properties ofsymmetric spaces and Bruhat�Tits buildings, Borel density, Mostow rigidity, geometricsuperrigidity, abstract and geometric arithmeticity, and residual �niteness of lattices.

Contents

1. Introduction 21.A. Notation 13

Part I. Structure theory of the isometry group 152. Convex subsets of the Tits boundary 152.A. Boundary subsets of small radius 152.B. Subspaces with boundary of large radius 162.C. Minimal actions and boundary-minimal spaces 173. Minimal invariant subspaces for subgroups 193.A. Existence of a minimal invariant subspace 193.B. Dichotomy 193.C. Normal subgroups 204. Algebraic and geometric product decompositions 214.A. Preliminary decomposition of the space 214.B. Proof of Theorem 1.1 and Addendum 1.4 234.C. CAT(0) spaces without Euclidean factor 245. Totally disconnected group actions 245.A. Smoothness 245.B. Locally �nite equivariant partitions and cellular decompositions 255.C. Alexandrov angle rigidity 265.D. On minimal normal subgroups of t.d. locally compact groups 285.E. Algebraic structure 326. Cocompact CAT(0) spaces 336.A. Fixed points at in�nity 336.B. Actions of simple algebraic groups 346.C. No branching geodesics 38

Date: August 8, 2008.Key words and phrases. Non-positive curvature, CAT(0) space, locally compact group, lattice.*Supported by IPDE and a Hodge fellowship.‡Supported in part by the Swiss National Science Foundation.

1

2 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONOD

6.D. No open stabiliser at in�nity 396.E. Cocompact stabilisers at in�nity 39

Part II. Discrete subgroups of the isometry group 427. An analogue of Borel density 427.A. Fixed points at in�nity 427.B. Geometric density for subgroups of �nite covolume 437.C. The limit set of subgroups of �nite covolume 458. CAT(0) lattices 468.A. Preliminaries on lattices 468.B. Variations on Auslander's theorem 478.C. Lattices, the Euclidean factor and �xed points at in�nity 488.D. Irreducible lattices in CAT(0) spaces 528.E. The hull of a lattice 548.F. On the canonical discrete kernel 558.G. Residually �nite lattices 569. CAT(0) superrigidity 579.A. CAT(0) superrigidity for some classical non-uniform lattices 579.B. CAT(0) superrigidity for irreducible lattices in products 589.C. Strong rigidity for product spaces 5910. Arithmeticity of abstract lattices 6010.A. Superrigid pairs 6110.B. Boundary maps 6410.C. Radical superrigidity 6510.D. Lattices with non-discrete commensurators 6610.E. Lattices in products of Lie and totally disconnected groups 6810.F. Lattices in general products 6911. Geometric arithmeticity 7211.A. CAT(0) lattices and parabolic isometries 7211.B. Arithmeticity of linear CAT(0) lattices 7411.C. A family of examples 7612. Some open questions 78References 79Index 84

1. Introduction

Non-positively curved metric spaces were introduced by A. D. Alexandrov [Ale57] andpopularised by M. Gromov, who called them CAT(0) spaces. They provide a framework forthe development of some form of generalised di�erential geometry, whose objects encompassRiemannian manifolds of non-positive sectional curvature as well as large families of singularspaces including Euclidean buildings and many other polyhedral complexes. Several aspectsof classical and less classical themes may thereby be approached in a uni�ed way. In orderto describe a few of these, we shall de�ne a CAT(0) lattice as a pair (Γ, X) consisting of aproper CAT(0) space X with cocompact isometry group and a lattice subgroup Γ < Is(X),i.e. a discrete subgroup of �nite invariant covolume (the compact-open topology makesIs(X) a locally compact second countable group which is thus canonically endowed with

ISOMETRY GROUPS OF NON-POSITIVELY CURVED SPACES 3

Haar measures). We say that (Γ, X) is uniform if Γ is cocompact in Is(X) or, equivalently,if the quotient Γ\X is compact; that case corresponds to Γ being a CAT(0) group in theusual terminology. Here are some examples.

� Let M be a compact Riemannian manifold of non-positive curvature and Γ itsfundamental group. The universal covering M is a CAT(0) space on which Γ actsproperly by deck transformations. The pair (Γ, M) is a uniform CAT(0) lattice.

� Many Gromov-hyperbolic groups Γ admit a properly discontinuous cocompact ac-tion on some CAT(−1) space X by isometries. In that case the pair (Γ, X) is auniform CAT(0) lattice. Amongst the examples arising in this way are hyperbolicCoxeter groups [Mou88], C ′(1

6) and C ′(14)-T (4) small cancellation groups [Wis04],

2-dimensional 7-systolic groups [J�06]. It is in fact a well known open problem ofM. Gromov to construct an example of a Gromov-hyperbolic group which is not aCAT(0) group (see [Gro93, 7.B]; also Remark 2.3(2) in Chapter III.Γ of [BH99]).

� An arithmetic group Γ may be realised as an irreducible lattice in a product Gof semi-simple algebraic group over local �elds. Moreover, any semi-simple groupover a local �eld possesses a canonical continuous, proper and cocompact actionon a symmetric space or a Bruhat�Tits building, according as the ground �eld isArchimedean or not. Thus G acts cocompactly on a product X of symmetric spacesand Bruhat�Tits buildings which, endowed with the L2-metric, is a proper CAT(0)space. Thus the pair (Γ, X) is a CAT(0) lattice, which we shall henceforth call anarithmetic CAT(0) lattice. More generally, any lattice in a semi-simple groupprovides an example of a CAT(0) lattice in a similar fashion.

� Extending some of the previous examples to a more general setting, tree latticesare CAT(0) lattices (Γ, X), where X is a locally �nite tree, see [BL01]. In [BM00b],striking examples of �nitely presented simple groups have been constructed asCAT(0) lattices whose underlying space is a product of two locally �nite trees.

� Aminimal adjoint Kac�Moody group Γ over a �nite �eld, as de�ned by J. Tits [Tit87],is endowed with two BN -pairs which yield strongly transitive Γ-actions on twobuildings denoted X+ and X− respectively. When the order of the ground �eld islarge enough, the pair (Γ, X+ ×X−) is a CAT(0) lattice.

Amongst these examples, the most important, and also the best understood, notablythrough the work of G. Margulis, consist undoubtedly of those arising from lattices in semi-simple groups over local �elds. It is therefore natural to address two sets of questions.

(a) What properties of these lattices are shared by all CAT(0) lattices?(b) What properties characterise them within the class of CAT(0) lattices?

In the same way as the study of arithmetic lattices requires a preliminary knowledge oftheir ambient semi-simple groups, tackling the above questions �rst necessitates a descrip-tion of the basic properties of the overall geometry of proper CAT(0) spaces and their fullisometry groups. Building upon earlier work which may be consulted in standard refer-ences including [BGS85], [Bal95] and [BH99], the �rst part of the present article providesa detailed analysis of the algebraic structure of the full isometry group of a proper CAT(0)space, independently of the existence of lattices. The second part is devoted to CAT(0)lattices and centres around the above questions.

We now proceed to describe our main results; for many of them, the core of the text willcontain a stronger, more precise and probably more cumbrous version.

.


Group decompositions. Let X be a proper CAT(0) space. Our �rst aim is to ob-tain structural results on the locally compact group Is(X) in a broad generality; we shallmostly ask that no point at in�nity be �xed simultaneously by all isometries of X. Thisnon-degeneracy assumption will be shown to hold automatically in the presence of lattices(Theorem 8.11 below).

Theorem 1.1. Let X be a proper CAT(0) space with �nite-dimensional Tits boundary.Assume that Is(X) has no global �xed point in ∂X.

Then there is a canonical closed convex Is(X)-stable subset X ′ ⊆ X such that G =Is(X ′) has a �nite index open characteristic subgroup G∗ � G which admits a canonicaldecomposition

(1.i) G∗ ∼= S1 × · · · × Sp ×(Rn o O(n)

)×D1 × · · · ×Dq (p, q, n ≥ 0)

where Si are almost connected simple Lie groups with trivial centre and Dj are totally dis-connected irreducible groups with trivial amenable radical. Any product decomposition of G∗

is a regrouping of the factors in (1.i).Moreover, all non-trivial normal, subnormal or ascending subgroups N < Dj are still

irreducible with trivial amenable radical and trivial centraliser in Dj. These properties alsohold for lattices in N and their normal, subnormal or ascending subgroups.

(A topological group is called irreducible if no �nite index open subgroup splits non-trivially as a direct product of closed subgroups. The amenable radical of a locallycompact group is the largest amenable normal subgroup; it is indeed a radical since theclass of amenable locally compact groups is stable under group extensions.)

Remarks 1.2.

(i) The �nite-dimensionality assumption holds automatically when X has a cocompactgroup of isometries [Kle99, Theorem C]. It is also automatic for uniquely geodesicspaces, e.g. manifolds (Proposition 6.11).

(ii) The conclusion fails in various ways if G �xes a point in ∂X.(iii) The quotient G/G∗ is just a group of permutations of possibly isomorphic factors

in the decomposition. In particular, G = G∗ oG/G∗.(iv) The canonical continuous homomorphism Is(X) → Is(X ′) = G is proper, but its

image sometimes has in�nite covolume.

.

. . . ísa tic �n eÒpoi sfaÐrac âgk¸mia, aÎt� tauta kaÈ fal�krac âgk¸mia diexèrqetai.

Sunèsioc KurenaÐou, Fal�krac âgk¸mion.1

Minimality. The de�nition of a CAT(0) space is �exible. For example, given a CAT(0)space X, there are many ways to deform it in order to construct another space Y , non-isometric toX, but with the property thatX and Y have isomorphic isometry groups or/andidentical boundaries. Amongst the simplest constructions, one can form (possibly warped)products with compact CAT(0) spaces or grow hair equivariantly along a discrete orbit.Much wilder (non-quasi-isometric) examples can be constructed for instance by considering

1Synesius of Cyrene, Fal�krac âgk¸mion (known as Calvitii encomium), end of Chapter 8 (at 72A inthe page numbering from Denis Pétau's 1633 edition). The Encomium was written around 402; we used the1834 edition by J. G. Krabinger (Ch. G. Lö�und, Stuttgart). The above excerpt translates roughly to: asmuch praise as is given to the spheres is due to the bald head too.


warped products with the very vast family of CAT(0) spaces having no isometries and aunique point at in�nity.

In order to address these issues, we introduce the following terminology.

An isometric action of a group G on a CAT(0) spaceX is said to beminimal if there is nonon-empty G-invariant closed convex subset X ′ ( X; the space X is itself calledminimal ifits full isometry group acts minimally. A CAT(0) space X is called boundary-minimal ifit possesses no closed convex subset Y ( X such that ∂Y = ∂X. Here is how these notionsrelate to one another.

Proposition 1.3. Let X be a proper CAT(0) space.

(i) Assume ∂X �nite-dimensional. If X is minimal, then it is boundary-minimal.(ii) Assume Is(X) has full limit set. If X is boundary-minimal, then it is minimal.(iii) If X is cocompact and geodesically complete, then it is both minimal and boundary-

minimal.

(Geodesic completeness means that every geodesic segment of can be extended to a bi-in�nite geodesic line � which need not be unique. This assumption is satis�ed by mostclassical examples, including [homology] manifolds and Euclidean buildings.)

In Theorem 1.1, the condition that Is(X) has no global �xed point at in�nity ensures theexistence of a closed convex Is(X)-invariant subset Y ⊆ X on which Is(X) acts minimally(see Proposition 3.1). The set of these minimal convex subsets possesses a canonical element,which is precisely the space X ′ which appears in Theorem 1.1.

.

De Rham decompositions. It is known that product decompositions of isometry groupsacting minimally and without global �xed point at in�nity induce a splitting of the space (forcocompact Hadamard manifolds, this is the Lawson�Yau [LY72] and Gromoll�Wolf [GW71]theorem; in general and for more references, see [Mon06]). It is therefore natural thatTheorem 1.1 is supplemented by a geometric statement.

Addendum 1.4. In Theorem 1.1, there is a canonical isometric decomposition

(1.ii) X ′ ∼= X1 × · · · ×Xp ×Rn × Y1 × · · · × Yqwhere G∗ acts componentwise according to (1.i) and G/G∗ permutes any isometric factors.All Xi and Yj are irreducible and minimal.

As it turns out, a geometric decomposition is the �rst of two independent steps in the proofof Theorem 1.1. In fact, we begin with an analogue of the de Rham decomposition [dR52]whose proof uses (a modi�cation of) arguments from the generalised de Rham theorem ofFoertsch�Lytchak [FL06]. In purely geometrical terms, we have the following statement.

Theorem 1.5. Let X be a proper boundary-minimal CAT(0) space with ∂X �nite-dimensional.Then X admits a canonical maximal isometric splitting

X ∼= Rn ×X1 × · · · ×Xm (n,m ≥ 0)

with each Xi irreducible and 6= R0,R1. Every isometry of X preserves this decompositionupon permuting possibly isometric factors Xi. Moreover, if X is minimal, so is every Xi.

To apply this theorem, it is desirable to know conditions ensuring boundary-minimality.In addition to the conditions provided by Proposition 1.3, we show that a canonical boundary-minimal subspace exists as soon as the boundary has circumradius > π/2 (Corollary 2.10).


In the second part of the proof of Theorem 1.1, we analyse the irreducible case whereX admits no isometric splitting, resulting in Theorem 1.6 to which we shall now turn.Combining these two steps, we then prove the unique decomposition of the groups, usingalso the splitting theorem from [Mon06].

.

Geometry of normal subgroups. In É. Cartan's correspondence between symmetricspaces and semi-simple Lie groups as well as in Bruhat�Tits theory, irreducible factors ofthe space correspond to simple groups. For general CAT(0) spaces and groups, simplicityfails of course very dramatically (free groups are perhaps the simplest, and yet most non-simple, CAT(0) groups). Nonetheless, we establish a geometric weakening of simplicity.

Theorem 1.6. Let X 6= R be an irreducible proper CAT(0) space with �nite-dimensionalTits boundary and G < Is(X) any subgroup whose action is minimal and does not have aglobal �xed point in ∂X.

Then every non-trivial normal subgroup N�G still acts minimally and without �xed pointin ∂X. Moreover, the amenable radical of N and the centraliser ZIs(G)(N) are both trivial;N does not split as a product.

This result can for instance be combined with the solution to Hilbert's �fth problem inorder to understand the connected component of the isometry group.

Corollary 1.7. Is(X) is either totally disconnected or an almost connected simple Lie groupwith trivial centre.

The same holds for any closed subgroup acting minimally and without �xed point at in-�nity.

A more elementary application of Theorem 1.6 uses the fact that elements with a discreteconjugacy class have open centraliser.

Corollary 1.8. If G is non-discrete, N cannot be a �nitely generated discrete subgroup.

A feature of Theorem 1.6 is that is can be iterated and thus applies to subnormal sub-groups. Recall that more generally a subgroup H < G is ascending if there is a (possiblytrans�nite) chain of normal subgroups starting with H and abutting to G. Using limitingarguments, we bootstrap Theorem 1.6 and show:

Theorem 1.9. Let N < G be any non-trivial ascending subgroup. Then all conclusions ofTheorem 1.6 hold for N .

.

Actions of simple algebraic groups. Both for the general theory and for the geometricsuperrigidity/arithmeticity statements of the second part of this paper, it is important tounderstand how algebraic groups act on CAT(0) spaces.

Theorem 1.10. Let k be a local �eld and G be an absolutely almost simple simply connectedk-group. Let X be a non-compact proper CAT(0) space on which G = G(k) acts continuouslyand cocompactly by isometries.

Then there is a G-equivariant bijection ∂X ∼= ∂Xmodel which is a homeomorphism withrespect to the cône topology and an isometry with respect to Tits' metric, where Xmodel isthe symmetric space or Bruhat�Tits building associated with G.

If in addition X is geodesically complete, then X is isometric to Xmodel.


A stronger and more detailed statement is provided below as Theorem 6.4; in particular,the cocompactness assumption can be relaxed, but we also show there by means of twoexamples that some assumptions remain necessary.

(As a point of terminology, we do not choose a particular scaling factor on symmetricspaces or Bruhat�Tits buildings, so that the isometries of our statement could be homoth-eties on each irreducible factor.)

.

Cocompact spaces. As already mentioned, a natural situation where the �nite-dimensionalityassumption in Theorem 1.1 holds automatically is when X has a cocompact group of isome-tries. This setting is particularly relevant in preparation to the study of CAT(0) groups andmore generally CAT(0) lattices. When Is(X) acts cocompactly, the space X is a tubularneighbourhood of X ′ and each factor in the decomposition (1.ii) of X ′ admits a cocompactgroup of isometries. As we shall see, this yields more precise information on the actionof both the connected and totally disconnected factors in the decomposition (1.i). Theconclusions become even stronger if X is assumed to be geodesically complete.

Theorem 1.11. Let X be a proper CAT(0) space whose isometry group acts cocompactlywithout �xed point at in�nity. Then all conclusions of Theorem 1.1 and Addendum 1.4 hold.Moreover:

If X has extensible geodesics, then X ′ = X, each Xi is isometric to the symmetric spaceof Gi and each group Dj acts by semi-simple isometries on Yj.

If X has uniquely extensible geodesics, then each group Dj is discrete.

Remark 1.12. It is not true in general that a minimal CAT(0) space is geodesically com-plete, even if one assumes that the isometry group acts cocompactly and without global�xed point at in�nity. Important examples of geodesically complete spaces are providedby Bruhat�Tits buildings and of course Hadamard manifolds, e.g. symmetric spaces. Acomplete CAT(0) space that is also a homology manifold has automatically extensiblegeodesics [BH99, II.5.12].

A further natural condition on non-discrete groups of isometries is that no open subgroup�xes a point at in�nity; this holds notably for symmetric spaces and Bruhat�Tits buildings.Although we shall not impose this condition in our other investigations, we show that muchstructure can be derived from it. For instance, on the geometric side, it turns out thatthe de Rham factors are either symmetric spaces or come quite close to form a reasonablecellular complex, as is the case for Bruhat�Tits buildings.

Theorem 1.13. Let X be a proper geodesically complete CAT(0) space such that someclosed subgroup G < Is(X) acts cocompactly. Suppose that no open subgroup of G �xes apoint at in�nity. Then X admits a canonical equivariant splitting

X ∼= X1 × · · · ×Xp × Y1 × · · · × Yqwhere each Xi is a symmetric space (possibly Euclidean) and each Yj possesses a G-equivariantlocally �nite decomposition into compact convex cells.

On the side of the group, we have the following result, for which we recall two de�nitions(which will be further discussed in the text). The quasi-centre consists of all elementswith open centraliser and the socle of a group is the subgroup generated by the (possiblyempty) family of all minimal non-trivial closed normal subgroups.


Theorem 1.14. In the setting of Theorem 1.13, assume that X has no Euclidean factor.Then:

(i) Every compact subgroup of G is contained in a maximal one; the maximal compactsubgroups fall into �nitely many conjugacy classes.

(ii) The quasi-centre of G is trivial; in particular G has no non-trivial discrete normalsubgroup.

(iii) The socle of G∗ is a direct product of p + q non-discrete characteristically simplegroups.

(iv) G possesses hyperbolic elements.

.

Smoothness. The rather common2 assumption of geodesic completeness has already oc-curred a few times above and is satis�ed by most classical examples of CAT(0) spaces(an outstanding counter-example being the convex core of Fuchsian groups). One of theimportant properties of geodesically complete spaces is that isometric actions of totallydisconnected groups are smooth in the following sense.

Theorem 1.15. Let X be a geodesically complete proper CAT(0) space X and G < Is(G)a totally disconnected (closed) subgroup acting minimally.

The the pointwise stabiliser in G of every bounded set is open.

This property, which is familiar from classical examples, fails in general (Remark 11.10).It is an important ingredient for statements such as Theorems 1.13 and 1.14 above as wellas for angle rigidity results regarding both the Alexandrov angle (Proposition 5.8) and theTits angle (Proposition 6.15).

.

A few cases of superrigidity. Combining the preceding general structure results withsome of Margulis' theorems, we obtain the following superrigidity statement.

Theorem 1.16. Let X be a proper CAT(0) space whose isometry group acts cocompactlyand without global �xed point at in�nity. Let Γ = SLn(Z) with n ≥ 3 and G = SLn(R).

For any isometric Γ-action on X there is a non-empty Γ-invariant closed convex subsetY ⊆ X on which the Γ-action extends uniquely to a continuous isometric action of G.

(The corresponding statement applies to all those lattices in semi-simple Lie groups thathave virtually bounded generation by unipotents.)

Observe that the above theorem has no assumptions whatsoever on the action; cocom-pactness is an assumption on the given CAT(0) space. It can happen that Γ �xes points in∂X, but its action on Y is without �xed points at in�nity and minimal (as we shall establishin the proof).

The assumption on bounded generation holds conjecturally for all non-uniform irreduciblelattices in higher rank semi-simple Lie groups (but always fails in rank one). It is known to

2One can always arti�cially make a CAT(0) space geodesically complete by gluing rays, though it is notalways possible to preserve properness (consider a compact but total set in an in�nite-dimensional Hilbertspace).


hold for arithmetic groups in split or quasi-split algebraic groups of a number �eld K of K-rank ≥ 2 by [Tav90], as well as in a few cases of isotropic but non-quasi-split groups [ER06];see also [WM07].

More generally, Theorem 1.16 holds for (S-)arithmetic groups provided the arithmeticsubgroup (given by integers at in�nite places) satis�es the above bounded generation prop-erty. For instance, the SLn example is as follows:

Theorem 1.17. Let X be a proper CAT(0) space whose isometry group acts cocompactlyand without global �xed point at in�nity. Let m be an integer with distinct prime factorsp1, . . . pk and set

Γ = SLn(Z[ 1m

]), G = SLn(R)× SLn(Qp1)× · · · × SLn(Qpk),

where n ≥ 3. Then for any isometric Γ-action on X there is a non-empty Γ-invariant closedconvex subset Y ⊆ X on which the Γ-action extends uniquely to a continuous isometricaction of G.

We point out that a �xed point property for similar groups acting on low-dimensionalCAT(0) cell complexes was established by B. Farb [Far08a].

Some of our general results also allow us to improve on the generality of the CAT(0) su-perrigidity theorem for irreducible lattices in arbitrary products of locally compact groupsproved in [Mon06]. For actions on proper CAT(0) spaces, the results of loc. cit. establishan unrestricted superrigidity on the boundary but require, in order to deduce superrigidityon the space itself, the assumption that the action be reduced (or alternatively �indecom-posable�).

We prove that, as soon as the boundary is �nite-dimensional, any action without global�xed point at in�nity is always reduced after suitably passing to subspaces and direct factors.It follows that the superrigidity theorem for arbitrary products holds in that generality, seeTheorem 9.4 below.

.

A �rst characterisation of symmetric spaces and Euclidean buildings. In sym-metric spaces and Bruhat�Tits buildings, the stabilisers of points at in�nity are exactlythe parabolic subgroups; as such, they are cocompact. This cocompactness holds furtherfor all Bass�Serre trees, namely bi-regular trees. Combining our results with results ofB. Leeb [Lee00] and A. Lytchak [Lyt05], we establish a corresponding characterisation.

Theorem 1.18. Let X be a geodesically complete proper CAT(0) space. Suppose that thestabiliser of every point at in�nity acts cocompactly on X.

Then X is isometric to a product of symmetric spaces, Euclidean buildings and Bass�Serretrees.

The Euclidean buildings appearing in the preceding statement admit an automorphismgroup that is strongly transitive, i.e. acts transitively on pairs (c, A) where c is a chamberand A an apartment containing c. This property characterises the Bruhat�Tits buildings,except perhaps for some two-dimensional cases where this is a known open question.

The above characterisation is of a di�erent nature and independent of the characterisa-tions using lattices that will be presented below.

.


Geometric density of lattices. As a transition between the general theory and the studyof CAT(0) lattices, we propose the following analogue of A. Borel's density theorem [Bor60].It also provides additional information about the totally disconnected groups Dj occurringin Theorem 1.1. It can of course be gainfully combined with Theorem 1.6.

Theorem 1.19. Let X be a proper CAT(0) space, G a locally compact group acting contin-uously by isometries on X and Γ < G a lattice. Suppose that X has no Euclidean factor.

If G acts minimally without �xed point at in�nity, so does Γ.

(This conclusion fails for spaces with a Euclidean factor. The theorem will be establishedmore generally for closed subgroups with �nite invariant covolume.)

As with classical Borel density, we shall use this density statement to derive statementsabout the centraliser, normaliser and radical of lattices in Section 7.

Remark 1.20. Theorem 1.19 applies to general proper CAT(0) spaces. Nonetheless, wepoint out that it implies in particular the classical Borel density theorem (see the end ofSection 7).

A more elementary variant of the above theorem shows that a large class of non-amenablegroups have rather restricted actions on proper CAT(0) spaces; as an application, one showsthat any isometric action of R. Thompson's group F on any proper CAT(0) space X has a�xed point in X, see Corollary 7.3.

.

Lattices: Euclidean factor, boundary, irreducibility and Mostow rigidity. Recallthat the Flat Torus theorem, originating in the work of Gromoll�Wolf [GW71] and Lawson�Yau [LY72], associates Euclidean subspaces Rn to any subgroup Zn of a CAT(0) group,see [BH99, �II.7]. (In the classical setting, when the CAT(0) group is given by a compactnon-positively curved manifold, this amounts to the seemingly more symmetric statementthat such a subgroup exists if and only if there is a �at torus is the manifold.)

The converse is a well known open problem stated by M. Gromov in [Gro93, �6.B3]; formanifolds see S.-T. Yau, problem 65 in [Yau82]). Point (i) in the following result is a (verypartial) answer; in the special case of cocompact Riemannian manifolds, this was the mainresult of P. Eberlein's article [Ebe83].

Theorem 1.21. Let X be a proper CAT(0) space, G < Is(X) a closed subgroup actingminimally and cocompactly on X and Γ < G a �nitely generated lattice. Then:

(i) If the Euclidean factor of X has dimension n, then Γ possesses a �nite index sub-group Γ0 which splits as Γ0 ' Zn × Γ′.Moreover, the dimension of the Euclidean factor is characterised as the maximalrank of a free Abelian normal subgroup of Γ.

(ii) G has no �xed point at in�nity; the set of Γ-�xed points at in�nity is contained inthe (possibly empty) boundary of the Euclidean factor.

The interest of point (ii) above is clear in view of all the preceding statements assumingthe absence of �xed points. In addition, it is already a �rst indication that the mere existenceof a (�nitely generated) lattice is a serious restriction on a proper CAT(0) space even withinthe class of cocompact minimal spaces. We recall that E. Heintze [Hei74] produced simplyconnected negatively curved Riemannian manifolds that are homogeneous (in particular,cocompact) but have a point at in�nity �xed by all isometries.


Since a CAT(0) lattice consists of a group and a space, there are two natural notions ofirreducibility: of the group or of the space. In the case of lattices in semi-simple groups, thetwo notions are known to coincide by a result of Margulis [Mar91, II.6.7]. We prove thatthis is the case for CAT(0) lattices as above.

Theorem 1.22. In the setting of Theorem 1.21, Γ is irreducible as an abstract group if andonly if for any �nite index subgroup Γ1 and any Γ1-equivariant decomposition X = X1×X2

with Xi non-compact, the projection of Γ1 to both Is(Xi) is non-discrete.

The combination of Theorem 1.22, Theorem 1.21 and of an appropriate form of superrigid-ity allow us to give a CAT(0) version of Mostow rigidity for reducible spaces (Section 9.C).

.

Geometric arithmeticity. We now expose results giving perhaps unexpectedly strongconclusions for CAT(0) lattices � both for the group and for the space. These results wereannounced in [CM08] in the case of CAT(0) groups; the present setting of �nitely generatedlattices is more general since CAT(0) groups are �nitely generated (cf. Lemma 8.3 below).

We recall that an isometry g is parabolic if the translation length infx∈X d(gx, x) is notachieved. For general CAT(0) spaces, parabolic isometries are not well understood; in fact,ruling out their existence can sometimes be the essential di�culty in rigidity statements.

Theorem 1.23. Let (Γ, X) be an irreducible �nitely generated CAT(0) lattice with Xgeodesically complete. Assume that X possesses some parabolic isometry.

If Γ is residually �nite, then X is a product of symmetric spaces and Bruhat�Tits build-ings. In particular, Γ is an arithmetic lattice unless X is a real or complex hyperbolic space.

If Γ is not residually �nite, then X still splits o� a symmetric space factor. Moreover,the �nite residual ΓD of Γ is in�nitely generated and Γ/ΓD is an arithmetic group.

(Recall that the �nite residual of a group is the intersection of all �nite index subgroups.)We single out a purely geometric consequence.

Corollary 1.24. Let (Γ, X) be a �nitely generated CAT(0) lattice with X geodesically com-plete.

Then X possesses a parabolic isometry if and only if X ∼= M×X ′, whereM is a symmetricspace of non-compact type.

The above results fail without the assumption of geodesic completeness, see Section 11.C.Nevertheless, we still obtain an arithmeticity statement when the underlying space admitssome parabolic isometry that is neutral, i.e. whose displacement length vanishes. Neutralparabolic isometries are even less understood, not even for their dynamical properties (whichcan be completely wild at least in Hilbert space [Ede64]); as for familiar examples, they areprovided by unipotent elements in semi-simple algebraic groups.

Theorem 1.25. Let (Γ, X) be an irreducible �nitely generated CAT(0) lattice. If X admitsany neutral parabolic isometry, then either:

(i) Is(X) is a rank one simple Lie group with trivial centre; or:(ii) Γ has a normal subgroup ΓD such that Γ/ΓD is an arithmetic group. Moreover, ΓD

is either �nite or in�nitely generated.

We turn to another type of statement of arithmeticity/geometric superrigidity. Havingestablished an abstract arithmeticity theorem (presented below as Theorem 1.29), we canappeal to our geometric results and prove the following.


Theorem 1.26. Let (Γ, X) be an irreducible �nitely generated CAT(0) lattice with Xgeodesically complete. Assume that Γ possesses some faithful �nite-dimensional linear rep-resentation (in characteristic 6= 2, 3).

If X is reducible, then Γ is an arithmetic lattice and X is a product of symmetric spacesand Bruhat�Tits buildings.

Section 11 contains more results of this nature but also demonstrates by a family ofexamples that some of the intricacies in the more detailed statements re�ect indeed theexistence of more exotic pairs (Γ, X).

.

Unique geodesic extension. Complete simply connected Riemannian manifolds of non-positive curvature, sometimes also called Hadamard manifolds, form a classical family ofproper CAT(0) spaces to which the preceding results may be applied. In fact, the naturalclass to consider in our context consists of those proper CAT(0) spaces in which every geo-desic segment extends uniquely to a bi-in�nite geodesic line. Clearly, this class contains allHadamard manifolds, but it presumably contains more examples. It is, however, somewhatrestricted with respect to the main thrust of the present work since it does not allow for,say, simplicial complexes; accordingly, the conclusions of the theorem below are also morestringent.

Theorem 1.27. Let X be a proper CAT(0) space with uniquely extensible geodesics. Assumethat Is(X) acts cocompactly without �xed points at in�nity.

(i) If X is irreducible, then either X is a symmetric space or Is(X) is discrete.(ii) If Is(X) possesses a �nitely generated non-uniform lattice Γ which is irreducible as

an abstract group, then X is a symmetric space (without Euclidean factor).(iii) Suppose that Is(X) possesses a �nitely generated lattice Γ (if Γ is uniform, this is

equivalent to the condition that Γ is a discrete cocompact group of isometries of X).If Γ is irreducible (as an abstract group) and X is reducible, then X is a symmetricspace (without Euclidean factor).

In the special case of Hadamard manifolds, statement (i) was known under the assumptionthat Is(X) satis�es the duality condition (without assuming that Is(X) acts cocompactlywithout �xed points at in�nity). This is due to P. Eberlein (Proposition 4.8 in [Ebe82]).Likewise, statement (iii) for manifolds is Proposition 4.5 in [Ebe82].

More recently, Farb�Weinberger [FW06] investigated analogous questions for asphericalmanifolds.

.

Groups without geometry. In the course of this work, we are led to establish somepurely group-theoretical results independently of any CAT(0) considerations; here are a fewexamples.

Let us agree that a topological group ismonolithic withmonolith L if the intersection Lof all non-trivial closed normal subgroups is itself non-trivial. If moreover the monolithis cocompact, then of course every (non-trivial) closed normal subgroup is cocompact, aproperty sometimes called �just non-compactness� in reference to just in�nite groups. Thecase of discrete groups of the following statement was established by J. Wilson [Wil71].


Proposition 1.28. Let G be a compactly generated non-compact locally compact group suchthat every non-trivial closed normal subgroup is cocompact.

Then either G is monolithic or discrete and residually �nite. In the former case, themonolith is either Rn or the direct product of �nitely many isomorphic topologically simplegroups.

We also study irreducible lattices in products of general topological groups in the abstractand establish the following arithmeticity statement.

Theorem 1.29. Let Γ < G = G1 × · · · × Gn be an irreducible �nitely generated lattice,where each Gi is any locally compact group.

If Γ admits a faithful Zariski-dense representation in a semi-simple group over some �eldof characteristic 6= 2, 3, then the amenable radical R of G is compact and the quasi-centreQZ (G) is virtually contained in Γ · R. Furthermore, upon replacing G by a �nite indexsubgroup, the quotient G/R splits as G+ ×QZ (G/R) where G+ is a semi-simple algebraicgroup and the image of Γ in G+ is an arithmetic lattice.

In shorter terms, this theorem states that up to a compact extension, G is the directproduct of a semi-simple algebraic group by a (possibly trivial) discrete group, and thatthe image of Γ in the non-discrete part is an arithmetic group. The assumption on thecharacteristic can be slightly weakened.

In the course of the proof, we characterise all irreducible �nitely generated lattices inproducts of the form G = S × D where S is a semi-simple Lie group and D a totallydisconnected group (Theorem 10.17). In particular, it turns our that D must necessarily belocally pro�nite by analytic. The corresponding question for simple algebraic groups insteadof Lie groups is also investigated (Theorem 10.19).

.

1.A. Notation. A metric space is proper if every closed ball is compact.We refer to Bridson and Hae�iger [BH99] for background on CAT(0) spaces. We recall

that the comparison angle ∠p(x, y) determined by three points p, x, y in any metric spaceis de�ned purely in terms of the corresponding three distances by looking at the correspond-ing Euclidean triangle. In other words, it is de�ned by

d2(x, y) = d2(p, x) + d2(p, y)− 2d(p, y)d(p, y) cos∠p(x, y).

The Alexandrov angle ∠p(x, y) in a CAT(0) space X is the non-increasing limit of thecomparison angle near p along the geodesic segments [p, x] and [p, y], see [BH99, II.3.1]. Inparticular, ∠p(x, y) ≤ ∠p(x, y). Likewise, geodesic rays from p determine the Alexandrovangle ∠p(ξ, η) for ξ, η ∈ ∂X. The Tits angle ∠T(ξ, η) is de�ned as the supremum of∠p(ξ, η) over all p ∈ X and has several useful characterisations given in Proposition II.9.8of [BH99].

Recall that to any point at in�nity ξ ∈ ∂X is associated the Busemann function

Bξ : X ×X → R : (x, y) 7→ Bξ,x(y)

de�ned by Bξ,x(y) = limt→∞(d(%(t), y) − d(%(t), x)), where % : [0,∞) → X is any geodesicray pointing towards ξ. The Busemann function does not depend on the choice of % and


satis�es the following:

Bξ,x(y) = −Bξ,y(x)Bξ,x(z) = Bξ,x(y) +Bξ,y(z) (the �cocycle relation�)Bξ,x(y) ≤ d(x, y).

Combining the de�nition of the Busemann function and of the comparison angle, we �ndthat if r is the geodesic ray pointing towards ξ with r(0) = x, then for any y 6= x we have

limt→∞

cos∠x(r(t), y) = −Bξ,x(y)d(x, y)

(the �asymptotic angle formula�).

By abuse of language, one refers to a Busemann function when it is more convenient toconsider the convex 1-Lipschitz function bξ : X → R de�ned by Bξ,x for some (usuallyimplicit) choice of base-point x ∈ X. We shall simply denote such a function by bξ in lowercase; they all di�er by a constant only in view of the cocycle relation.

The boundary at in�nity ∂X is endowed with the cône topology [BH99, II.8.6] as wellas with the (much �ner) topology de�ned by the Tits angle. The former is often implicitlyunderstood, but when referring to dimension or radius, the topology and distance de�nedby the Tits angle are considered (this is sometimes emphasised by referring to the �Titsboundary�). The later distance is not to be confused with the associated length metric called�Tits distance� in the literature; we will not need this concept (except in the discussions atthe beginning of Section 6).

Recall that any complete CAT(0) space splits o� a canonical maximal Hilbertian factor(Euclidean in the proper case studied here) and any isometry decomposes accordingly, seeTheorem II.6.15(6) in [BH99].

Normalisers and centralisers in a group G are respectively denoted by NG and ZG. Whensome group G acts on a set and x is a member of this set, the stabiliser of x in G is denotedby StabG(x) or by the shorthand Gx. For the notation regarding algebraic groups, we followthe standard notation as in [Mar91].

Finally, we present two remarks that will never be used below but give some context oncertain frequent assumptions.

The �rst remark is the following CAT(0) version of the Hopf�Rinow theorem: everygeodesically complete locally compact CAT(0) space is proper. Surprisingly, we could not�nd this statement in the literature (though a di�erent statement is often referred to as theHopf�Rinow theorem, see [BH99, I.3.7]). As pointed out orally by A. Lytchak, the aboveresult is readily established by following the strategy of proof of [BH99, I.3.7] and extendinggeodesics.

The second fact is that if a proper CAT(0) space is �nite-dimensional (in the senseof [Kle99]), then so is its Tits boundary (generalising for instance Proposition 6.11 below).The argument (which will be found in [CL]) goes as follows. For any sphere S in thespace X, the �visual map� ∂X → S is Tits-continuous; if it were injective, the result wouldfollow. However, it becomes injective after replacing S with the ultraproduct of spheresof unbounded radius by the very de�nition of the boundary; the ultraproduct constructionpreserves the bound on the dimension, �nishing the proof.


Part I. Structure theory of the isometry group

2. Convex subsets of the Tits boundary

2.A. Boundary subsets of small radius. Given a metric space X and a subset Z ⊆ X,one de�nes the circumradius of Z in X as

infx∈X

supz∈Z

d(x, z).

A point x realising the in�mum is called a circumcentre of Z in X. The intrinsic cir-cumradius of Z is its circumradius in Z itself; one de�nes similarly an intrinsic circum-centre. It is called canonical if it is �xed by every isometry of X which stabilises Z. Weshall make frequent use of the following construction of circumcentres, due to A. Balser andA. Lytchak [BL05, Proposition 1.4]:

Proposition 2.1. Let X be a complete CAT(1) space and Y ⊆ X be a �nite-dimensionalclosed convex subset. If Y has intrinsic circumradius ≤ π/2, then the set C(Y ) of intrinsiccircumcentres of Y has a unique circumcentre, which is therefore a canonical (intrinsic)circumcentre of Y . �

Let now X be a proper CAT(0) space.

Proposition 2.2. Let X0 ⊃ X1 ⊃ . . . be a nested sequence of non-empty closed convexsubsets of X such that

⋂nXn is empty. Then the intersection

⋂n ∂Xn is a non-empty

closed convex subset of ∂X of intrinsic circumradius at most π/2.In particular, if the Tits boundary is �nite-dimensional, then

⋂n ∂Xn has a canonical

intrinsic circumcentre.

Proof. Pick any x ∈ X and let xn be its projection to Xn. The assumption⋂nXn = ∅

implies that xn goes to in�nity. Upon extracting, we can assume that it converges tosome point ξ ∈ ∂X; observe that ξ ∈

⋂n ∂Xn. We claim that any η ∈

⋂n ∂Xn satis�es

∠T(ξ, η) ≤ π/2. The proposition then follows because (i) the boundary of any closed convexset is closed and π-convex [BH99, II.9.13] and (ii) each ∂Xn is non-empty since otherwiseXn would be bounded, contradicting

⋂nXn = ∅. When ∂X has �nite dimension, there is

a canonical intrinsic circumcentre by Proposition 2.1.For the claim, observe that there exists a sequence of points yn ∈ Xn converging to

η. It su�ces to prove that the comparison angle ∠x(xn, yn) is bounded by π/2 for all n,see [BH99, II.9.16]. This follows from

∠xn(x, yn) ≥ ∠xn(x, yn) ≥ π/2,where the second inequality holds by the properties of the projection on a convex set [BH99,II.2.4(3)]. �

The combination of the preceding two propositions has the following consequence, whichimproves the results established by Fujiwara, Nagano and Shioya (Theorems 1.1 and 1.3in [FNS06]).

Corollary 2.3. Let g be a parabolic isometry of X. The following assertions hold:

(i) The �xed point set of g in ∂X has intrinsic circumradius at most π/2.(ii) If ∂X �nite-dimensional, then the centraliser ZIs(X)(g) has a canonical global �xed

point in ∂X.(iii) For any subgroup H < Is(X) containing g, the (possibly empty) �xed point set of

H in ∂X has circumradius at most π/2. �


Here is another immediate consequence.

Corollary 2.4. Let G be a topological group with a continuous action by isometries on Xwithout global �xed point. Suppose that G is the union of an increasing sequence of compactsubgroups and that ∂X is �nite-dimensional. Then there is a canonical G-�xed point in ∂X,�xed by all isometries normalising G.

Proof. Consider the sequence of �xed point sets XKn of the compact subgroups Kn. Itsintersection is empty by assumption and thus Proposition 2.2 applies. �

Finally, we record the following elementary fact, which may also be deduced by means ofProposition 2.2:

Lemma 2.5. Let ξ ∈ ∂X. Given any closed horoball B centred at ξ, the boundary ∂Bcoincides with the ball of Tits radius π/2 centred at ξ in ∂X.

Proof. Any two horoballs centred at the same point at in�nity lie at bounded Hausdor� dis-tance from one another. Therefore, they have the same boundary at in�nity. In particular,the boundary ∂B of the given horoball coincides with the intersection of the boundaries ofall horoballs centred at ξ. By Proposition 2.2, this is of circumradius at most π/2; in factthe proof of that proposition shows precisely that the set is contained in the ball of radiusat most π/2 around ξ.

Conversely, let η ∈ ∂X be a point which does not belong to ∂B. We claim that ∠T(ξ, η) ≥π/2. This shows that every point of ∂X at Tits distance less than π/2 from ξ belongs to∂B. Since the latter is closed, it follows that ∂B contains the closed ball of Tits radius π/2

We turn to the claim. Let bξ be a Busemann function centred at ξ. Since every geodesicray pointing towards η escapes every horoball centred at ξ, there exists a ray % : [0,∞)→ Xpointing to η such that bξ(%(0)) = 0 and bξ(%(t)) > 0 for all t > 0 (actually, this increasesto in�nity by convexity). Let c : [0,∞)→ X be the geodesic ray emanating from %(0) andpointing to ξ. We have ∠T(ξ, η) = limt,s→∞∠%(0)(%(t), c(s)), see [BH99, II.9.8]. Thereforethe claim follows from the asymptotic angle formula (Section 1.A) by taking y = c(s) withs large enough. �

2.B. Subspaces with boundary of large radius. As before, let X be a proper CAT(0)space. The following result improves Proposition 2.2 in [Lee00]:

Proposition 2.6. Let Y ⊆ X be a closed convex subset such that ∂Y has intrinsic cir-cumradius > π/2. Then there exists a closed convex subset Z ⊆ X with ∂Z = ∂Y whichis minimal for these properties. Moreover, the union Z0 of all such minimal subspaces isclosed, convex and splits as a product Z0

∼= Z × Z ′.

Proof. If no minimal such Z existed, there would be a chain of such subsets with emptyintersection. The distance to a base-point must then go to in�nity and thus the chain con-tains a countable sequence to which we apply Proposition 2.2, contradicting the assumptionon the circumradius.

Let Z ′ denote the set of all such minimal sets and Z0 =⋃Z ′ be its union. As in [Lee00,

p. 10] one observes that for any Z1, Z2 ∈ Z ′, the distance z 7→ d(z, Z2) is constant on Z1

and that the nearest point projection pZ2 restricted to Z1 de�nes an isometry Z1 → Z2.By the Sandwich Lemma [BH99, II.2.12], this implies that Z0 is convex and that the mapZ ′×Z ′ → R+ : (Z1, Z2) 7→ d(Z1, Z2) is a geodesic metric on Z ′. As in [Mon06, Section 4.3],this yields a bijection α : Z0 → Z×Z ′ : x 7→ (pZ(x), Zx), where Zx is the unique element ofZ ′ containing x. The product of metric spaces Z ×Z ′ is given the product metric. In order


to establish that α is an isometry, it remains as in [Mon06, Proposition 38], to trivialise�holonomy�; it the current setting, this is achieved by Lemma 2.7, which thus concludes theproof of Proposition 2.6. (Notice that Z0 is indeed closed since otherwise we could extendα−1 to the completion of Z × Z ′.) �

Lemma 2.7. For all Z1, Z2, Z3 ∈ Z ′, we have pZ1 ◦ pZ3 ◦ pZ2 |Z1 = IdZ1.

Proof of Lemma 2.7. Let ϑ : Z1 → Z1 be the isometry de�ned by pZ1 ◦ pZ3 ◦ pZ2 |Z1 and letf be its displacement function. Then f : Z1 → R is a non-negative convex function whichis bounded above by d(Z1, Z2) + d(Z2, Z3) + d(Z3, Z1). In particular, the restriction of f toany geodesic ray in Z1 is non-increasing. Therefore, a sublevel set of f is a closed convexsubset Z of Z1 with full boundary, namely ∂Z = ∂Z1. By de�nition, the subspace Z1 isminimal with respect to the property that ∂Z1 = ∂Y and hence we deduce Z = Z1. Itfollows that the convex function f is constant. In other words, the isometry ϑ is a Cli�ordtranslation. If it is not trivial, then Z1 would contain a ϑ-stable geodesic line on which ϑacts by translation. But by [BH99, Lemma II.2.15], the restriction of ϑ to any geodesic lineis the identity. Therefore ϑ is trivial, as desired. �

Let Γ be a group acting on X by isometries. Following [Mon06, De�nition 5], we say thatthe Γ-action is reduced if there is no unbounded closed convex subset Y ( X such thatg.Y is at �nite Hausdor� distance from Y for all g ∈ Γ.

Corollary 2.8. Let X be a proper irreducible CAT(0) space with �nite-dimensional Titsboundary, and Γ < Is(X) be a subgroup acting minimally without �xed point at in�nity.Then the Γ-action is reduced.

Proof. Suppose for a contradiction that the Γ-action on X is not reduced. Then there existsan unbounded closed convex subset Y ( X such that g.Y is at �nite Hausdor� distancefrom Y for all g ∈ Γ. In particular ∂Y is Γ-invariant. By Proposition 2.1, it must haveintrinsic circumradius > π/2. Proposition 2.6 therefore yields a canonical closed convexsubset Z0 = Z × Z ′ with ∂(Z × {z′}) = ∂Y for all z′ ∈ Z ′; clearly Z0 is Γ-invariant andhence we have Z0 = X by minimality. Since X is irreducible by assumption, we deduceX = Z and hence X = Y , as desired. �

2.C. Minimal actions and boundary-minimal spaces. Boundary-minimality and min-imality, as de�ned in the Introduction, are two possible ways for a CAT(0) space to be�non-degenerate�, as illustrated by the following.

Lemma 2.9. Let X be a complete CAT(0) space.

(i) A group G < Is(X) acts minimally if and only if any continuous convex G-invariantfunction on X is constant.

(ii) If X is boundary-minimal then any bounded convex function on X is constant.

Proof. Necessity in the �rst assertion follows immediately by considering sub-level sets(see [Mon06, Lemma 37]). Su�ciency is due to the fact that the distance to a closed convexset is a convex continuous function [BH99, II.2.5]. The second assertion was established inthe proof of Lemma 2.7. �

Proposition 2.6 has the following important consequence:

Corollary 2.10. Let X be a proper CAT(0) space. If ∂X has circumradius > π/2, then Xpossesses a canonical closed convex subspace Y ⊆ X such that Y is boundary-minimal and∂Y = ∂X.


Proof. Let Z0 = Z × Z ′ be the product decomposition provided by Proposition 2.6. Thegroup Is(X) permutes the elements of Z ′ and hence acts by isometries on Z ′. Under thepresent hypotheses, the space Z ′ is bounded since ∂Z = ∂X. Therefore it has a circumcentrez′, and the �bre Y = Z × {z′} is thus Is(X)-invariant. �

Proposition 2.11. Let X be a proper CAT(0) space which is minimal. Assume that ∂Xhas �nite dimension. Then ∂X has circumradius > π/2 (unless X is reduced to a point).In particular, X is boundary-minimal.

The proof of Proposition 2.11 requires some preliminaries. Given a point at in�nity ξ,consider the Busemann function Bξ; the cocycle property (recalled in Section 1.A) impliesin particular that for any isometry g ∈ Is(X) �xing ξ and any x ∈ X the real numberBξ,x(g.x) is independent on the choice of x and yields a canonical homomorphism

βξ : Is(X)ξ −→ R : g 7−→ Bξ,x(g.x)

called the Busemann character centred at ξ.

Given an isometry g, it follows by the CAT(0) property that infn≥0 d(gnx, x)/n coincideswith the translation length of g independently of x. We call an isometry ballistic whenthis number is positive. An important fact about a ballistic isometry g of any completeCAT(0) space X is that for any x ∈ X the sequence {gn.x}n≥0 converges to a point ηg ∈ ∂Xindependent of x; ηg is called the (canonical) attracting �xed point of g in ∂X. Moreover,this convergence holds also in angle, which means that lim∠x(gnx, r(t)) vanishes as n, t→∞when r : R+ → X is any ray pointing to ηg. This is a (very) special case of the resultsin [KM99].

Lemma 2.12. Let ξ ∈ X and g ∈ Is(X)ξ be an isometry which is not annihilated by theBusemann character centred at ξ. Then g is ballistic. Furthermore, if βξ(g) > 0 then∠T(ξ, ηg) > π/2.

Proof. We have βξ(g) = Bξ,x(g.x) ≤ d(x, g.x) for all x ∈ X. Thus g is ballistic as soon asβξ(g) is non-zero.

Assume βξ(g) > 0 and suppose for a contradiction that ∠T(ξ, ηg) ≤ π/2. Choose x ∈ Xand let %, σ be the rays issuing from x and pointing towards ξ and ηg respectively. Recallfrom [BH99, II.9.8] that ∠T(ξ, ηg) = limt,s→∞∠x(%(t), σ(s)). The convergence in directionof gnx implies that this angle is also given by limt,n→∞∠x(%(t), gnx). Since βξ(g) > 0 wecan �x n large enough to have

cos lim inft→∞

∠x(%(t), gnx) > −βξ(g)d(gx, x)

.

We now apply the asymptotic angle formula from Section 1.A with y = gnx and deducethat the left hand side is −βξ(gnx)/d(gnx, x). Since βξ(gnx) = nβξ(gx) and d(gnx, x) ≤nd(gx, x), we have a contradiction. �

Proof of Proposition 2.11. We can assume that ∂X is non-empty since otherwise X is apoint by minimality. Suppose for a contradiction that its circumradius is ≤ π/2. Then Is(X)possesses a global �xed point ξ ∈ ∂X and ξ is a circumcentre of ∂X, see Proposition 2.1.Lemma 2.12 implies that Is(X) = Is(X)ξ is annihilated by the Busemann character centredat ξ. Thus Is(X) stabilises every horoball, contradicting minimality. �

We shall use repeatedly the following elementary fact.


Lemma 2.13. Let G be a group with an isometric action on a proper geodesically completeCAT(0) space X. If G acts cocompactly or more generally has full limit set, then the actionis minimal. (This holds more generally when ∆G = ∂X in the sense of Section 3.B below.)

Proof. Let Y ⊆ X be a a non-empty closed convex invariant subset, choose y ∈ Y andsuppose for a contradiction that there is x /∈ Y . Let r : R+ → X be a geodesic ray startingat y and going through x. By convexity [BH99, II.2.5(1)], the function d(r(t), Y ) tends toin�nity and thus r(∞) /∈ ∂Y . This is absurd since ∆G ⊆ ∂Y . �

Proof of Proposition 1.3. (i) See Proposition 2.11.

(ii) Since Is(X) has full limit set, any Is(X)-invariant subspace has full boundary. Minimalityfollows, since boundary-minimality ensures that X possesses no proper subspace with fullboundary.

(iii) X is minimal by Lemma 2.13, hence boundary-minimal by (i), since any cocompactspace has �nite-dimensional boundary by [Kle99, Theorem C]. �

3. Minimal invariant subspaces for subgroups

3.A. Existence of a minimal invariant subspace. For the record, we recall the followingelementary dichotomy; a re�nement will be given in Theorem 3.3 below:

Proposition 3.1. Let G be a group acting by isometries on a proper CAT(0). Then eitherG has a global �xed point at in�nity, or any �ltering family of non-empty closed convexG-invariant subsets has non-empty intersection.

(Recall that a family of sets is �ltering if it is directed by containment ⊇.)

Proof. (Remark 36 in [Mon06].) Suppose Y is such a family, choose x ∈ X and let xYbe its projection on each Y ∈ Y . If the net {xY }Y ∈Y is bounded, then

⋂Y ∈Y Y is non-

empty. Otherwise it goes to in�nity and any accumulation point in ∂X is G-�xed in viewof d(gxY , xY ) ≤ d(gx, x). �

3.B. Dichotomy. Let G be a group acting by isometries on a complete CAT(0) space X.

Lemma 3.2. Given any two x, y ∈ X, the convex closures of the respective G-orbits of xand y in X have the same boundary in ∂X.

Proof. Let Y be the convex closure of the G-orbit of x. In particular Y is the minimal closedconvex G-invariant subset containing x. Given any closed convex G-invariant subset Z, letr = d(x, Z). Recall that the tubular closed neighbourhoodNr(Z) is convex [BH99, II.2.5(1)].Since it is also G-invariant and contains x, the minimality of Y implies Y ⊆ Nr(Z). �

This yields a canonical closed convex G-invariant subset of the boundary ∂X, which wedenote by ∆G. It contains the limit set ΛG but is sometimes larger.

Combining what we established thus far with the splitting arguments from [Mon06], weobtain a dichotomy:

Theorem 3.3. Let G be a group acting by isometries on a complete CAT(0) space X andH < G any subgroup.If H admits no minimal non-empty closed convex invariant subset and X is proper, then:

(A.i) ∆H is a non-empty closed convex subset of ∂X of intrinsic circumradius at most π/2.(A.ii) If ∂X is �nite-dimensional, then the normaliser NG(H) of H in G has a global

�xed point in ∂X.


If H admits a minimal non-empty closed convex invariant subset Y ⊆ X, then:

(B.i) The union Z of all such subsets is a closed convex NG(H)-invariant subset.(B.ii) Z splits H-equivariantly and isometrically as a product Z ' Y × C, where C is a

complete CAT(0) space which admits a canonical NG(H)/H-action by isometries.(B.iii) If the H-action on X is non-evanescent, then C is bounded and there is a canonical

minimal non-empty closed convex H-invariant subset which is NG(H)-stable.

(When X is proper, the non-evanescence condition of (iii) simply means that H has no�xed point in ∂X; see [Mon06].)

Proof. In view of Lemma 3.2, the set ∆H is contained in the boundary of any non-emptyclosed convex H-invariant set and is NG(H)-invariant. Thus the assertions (A.i) and (A.ii)follow from Proposition 2.2, noticing that in a proper space ∆H is non-empty unless H hasbounded orbits, in which case it �xes a point, providing a minimal subspace. For (B.i),(B.ii) and (B.iii), see Remarks 39 in [Mon06]. �

3.C. Normal subgroups.

Proof of Theorem 1.6. We adopt the notation and assumptions of the theorem. By (A.ii), Nadmits a minimal non-empty closed convex invariant subset Y ⊆ X. This set is unbounded,since otherwise N �xes a point and thus by G-minimality XN = X, hence N = 1. Since Xis irreducible, points (B.i) and (B.ii) show Y = X and thus N acts indeed minimally.

Since the displacement function of any g ∈ ZG(N) is a convex N -invariant function, it isconstant by minimality. Hence g is a Cli�ord translation and must be trivial since otherwiseX splits o� a Euclidean factor, see [BH99, II.6.15].

The derived subgroup N ′ = [N,N ] is also normal in G and therefore acts minimally bythe previous discussion, noticing that N ′ is non-trivial since otherwise N ⊆ ZG(N). IfN �xed a point at in�nity, N ′ would preserve all corresponding horoballs, contradictingminimality.

Having established that N acts minimally and without �xed point at in�nity, we canapply the splitting theorem (Corollary 10 in [Mon06]) and deduce from the irreducibility ofX that N does not split.

Finally, let R � N be the amenable radical and observe that it is normal in G. Thetheorem of Adams�Ballmann [AB98a] states that R either (i) �xes a point at in�nity or(ii) preserves a Euclidean �at in X. (Although their result is stated for amenable groupswithout mentioning any topology, the proof applies indeed to every topological group thatpreserves a probability measure whenever it acts continuously on a compact metrisablespace.) If R is non-trivial, we know already from the above discussion that (i) is impossibleand that R acts minimally; it follows that X is a �at. By irreducibility and since X 6= R,this forces X to be a point, contradicting R 6= 1. �

Corollary 1.7 will be proved in Section 4.B. For Corollary 1.8, it su�ces to observe thatthe centraliser of any element of a discrete normal subgroup is open. Next, we recall thefollowing de�nition.

A subgroup N of a group G is ascending if there is a family of subgroups Nα < Gindexed by the ordinals and such that N0 = N , Nα �Nα+1, Nα =

⋃β<αNβ if α is a limit

ordinal and Nα = G for α large enough. The smallest such ordinal is the order.

Proposition 3.4. Consider a group acting minimally by isometries on a proper CAT(0)space. Then any ascending subgroup without global �xed point at in�nity still acts minimally.


Proof. We argue by trans�nite induction on the order ϑ of ascending subgroups N < G, thecase ϑ = 0 being trivial. Let X be a space as in the statement. By Proposition 3.1, eachNα has a minimal set. If ϑ = ϑ′ + 1, it follows from (B.iii) that Nϑ′ acts minimally andwe are done by induction hypothesis. Assume now that ϑ is a limit ordinal. For all α, wedenote as in (B.i) by Zα ⊆ X the union of all Nα-minimal sets. The induction hypothesisimplies that for all α ≤ β < ϑ, any Nβ-minimal set is Nα-minimal. Thus, if Z0 = Y0×C0 isa splitting as in (B.ii) with a N -minimal set Y0, we have a nested family of decompositionsZα = Y0 × Cα for a nested family of closed convex subspaces Cα of the compact CAT(0)space C0, indexed by α < ϑ. Thus, for any c ∈

⋂α<ϑCα, the space Y0 × {c} is G-invariant

and hence Y0 = X indeed. �

Remark 3.5. Proposition 3.4 holds more generally for complete CAT(0) spaces if N isnon-evanescent. Indeed Proposition 3.1 hold in that generality (Remark 36 in [Mon06]) andC remains compact in a weaker topology (Theorem 14 in [Mon06]).

Proof of Theorem 1.9. In view of Theorem 1.6, it su�ces to prove that any non-trivialascending subgroup N < G as in that statement still acts minimally and without global�xed point at in�nity. We argue by induction on the order ϑ and we can assume that ϑ is alimit ordinal by Theorem 1.6. Then

⋂α<ϑ(∂X)Nα is empty and thus by compactness there

is some α < ϑ such that (∂X)Nα is empty. Now Nα acts minimally on X by Proposition 3.4and thus we conclude using the induction hypothesis. �

4. Algebraic and geometric product decompositions

4.A. Preliminary decomposition of the space. We shall prepare our spaces by means ofa geometric decomposition. For any geodesic metric space with �nite a�ne rank, Foertsch�Lytchak [FL06] established a canonical decomposition generalising the classical theoremof de Rham [dR52]. However, such a statement fails to be true for CAT(0) spaces thatare merely proper, due notably to compact factors that can be in�nite products. Never-theless, using asymptotic CAT(0) geometry and Section 2.A, we can adapt the argumentsfrom [FL06] and obtain:

Theorem 4.1. Let X be a proper CAT(0) space with ∂X �nite-dimensional and of cir-cumradius > π/2. Then there is a canonical closed convex subset Z ⊆ X with ∂Z = ∂X,invariant under all isometries, and admitting a canonical maximal isometric splitting

(4.i) Z ∼= Rn × Z1 × · · · × Zm (n,m ≥ 0)

with each Zi irreducible and 6= R0,R1. Every isometry of Z preserves this decompositionupon permuting possibly isometric factors Zi.

Remark 4.2. It is well known that in the above situation the splitting (4.i) induces adecomposition

Is(Z) = Is(Rn)×((

Is(Z1)× · · · × Is(Zm))

o F),

where F is the permutation group of {1, . . . , d} permuting possible isometric factors amongstthe Yj . Indeed, this follows from the statement that isometries preserve the splitting uponpermutation of factors, see e.g. Proposition I.5.3(4) in [BH99]. Of course, this does not apriori mean that we have a unique, nor even canonical, splitting in the category of groups;this shall however be established for Theorem 1.1.

The hypotheses of Theorem 4.1 are satis�ed in some naturally occurring situations:

Corollary 4.3. Let X be a proper CAT(0) space with �nite-dimensional boundary.


(i) If Is(X) has no �xed point at in�nity, then X possesses a subspace Z satisfying allthe conclusions of Theorem 4.1.

(ii) If Is(X) acts minimally, then X admits a canonical splitting as in 4.i.

Proof of Corollary 4.3. By Proposition 2.1, if Is(X) has no �xed point at in�nity, then ∂Xhas circumradius > π/2. By Proposition 2.11, the same conclusion holds is Is(X) actsminimally. �

Proof of Theorems 1.5 and 4.1. For Theorem 4.1, we let Z ⊆ X be the canonical boundary-minimal subset with ∂Z = ∂X provided by Corollary 2.10; we shall not use the circumradiusassumption any more. For Theorem 1.5, we let Z = X. The remainder of the argument iscommon for both statements.

Recalling that in complete generality all isometries preserving the Euclidean factor de-composition [BH99, II.6.15], we can assume that Z has no Euclidean factor and shall obtainthe decomposition (4.i) with n = 0.

Since Z is minimal amongst closed convex subsets with ∂Z = ∂X, it has no non-trivialcompact factor. On the other hand, any proper geodesic metric space admits some maximalproduct decomposition into non-compact factors. In conclusion, Z admits some maximalsplitting Z = Z1×· · ·×Zm with each Zi irreducible and 6= R0,R1. (This can fail in presenceof compact factors).

It remains to prove that any other such decomposition Z = Z ′1×· · ·×Z ′m′ coincides withthe �rst one after possibly permuting the factors (in particular, m′ = m). We now borrowfrom the argumentation in [FL06], indicating the steps and the necessary changes. It isassumed that the reader has a copy of [FL06] at hand but keeps in mind that our spacesmight lack the �nite a�ne rank condition assumed in that paper. We shall replace thenotion of a�ne subspaces with a large-scale particular case: a cône shall be any subspaceisometric to a closed convex cône in some Euclidean space. This includes the particularcases of a point, a ray or a full Euclidean space.

Whenever a space Y has some product decomposition and Y ′ is a factor, write Y ′y ⊆ Yfor the corresponding �bre Y ′y ∼= Y ′ through y ∈ Y . The following is an analogue ofCorollary 1.2 in [FL06].

Lemma 4.4. Let Y be a proper CAT(0) space with �nite-dimensional boundary and withoutcompact factors. Suppose given two decompositions Y = Y1 × Y2 = S1 × S2 with all four(Yi)y ∩ (Sj)y reduced to {y} for some y ∈ Y . Then Y is a Euclidean space.

Proof of Lemma 4.4. Any y ∈ Y is contained in a maximal cône based at y since ∂Y has�nite dimension; by abuse of language we call such cônes maximal. The arguments ofSections 3 and 4 in [FL06] show that any maximal cône is rectangular, which means that itinherit a product structure from any product decomposition of the ambient CAT(0) space.Speci�cally, it su�ces to observe that the product of two cônes is a cône and that theprojection of a cône along a product decomposition of CAT(0) spaces remains a cône. (Infact, the �equality of slopes� of Section 4.2 in [FL06], namely the fact that parallel geodesicsegments in a CAT(0) space have identical slopes in product decompositions, is a generalfact for CAT(0) spaces. It follows from the convexity of the metric, see for instance [Mon06,Proposition 49] for a more general statement.) The deduction of the statement of Lemma 4.4from the rectangularity of maximal cônes following [FL06] is particularly short since allproper CAT(0) Banach spaces are Euclidean. �

Lemma 4.5. For a given z ∈ Z and any product decomposition Z = S×S′, the intersection(Zi)z ∩ Sz is either {z} or (Zi)z.


Proof of Lemma 4.5. Write PS : Z → S and PZi : Z → Zi for the projections and setFz = Sz ∩ (Zi)z. Following [FL06], de�ne T ⊆ Z by T = PS(Fz) × S′. We contend thatPZi(T ) has full boundary in Zi.

Indeed, given any point in ∂(Zi)z, we represent is by a ray r originating from z. We canchoose a maximal cône in Z based at z and containing r. We know already that this côneis rectangular, and therefore the proof of Lemma 5.2 in [FL06] shows that PZi(r) lies inPZi(T ), justifying our contention.

We observe that Zi inherits from Z the property that it has no closed convex propersubset of full boundary. In conclusion, since PZi(T ) is a convex set, it is dense in Z.However, according to Lemma 5.1 in [FL06], it splits as PZi(T ) = PZi(Fz)×PZi(S′). Uponpossibly replacing PZi(S′) by its completion (whilst PZi(Fz) is already closed in Zi sincePZi is isometric on (Zi)z), we obtain a splitting of the closure of PZi(T ), and hence of Zi.This completes the proof of the lemma since Zi is irreducible. �

Now the main argument runs by induction overm ≥ 2. Lemma 4.5 identi�es by inductionZi with some Z ′j . Indeed, Lemma 4.4 excludes that all pairwise intersections reduce to apoint since Z has no Euclidean factor. �

4.B. Proof of Theorem 1.1 and Addendum 1.4. The following consequence of thesolution to Hilbert's �fth problem belongs to the mathematical lore.

Theorem 4.6. Let G be a locally compact group with trivial amenable radical. Then Gpossesses a canonical �nite index open normal subgroup G† such that G† = L×D, where Lis a connected semi-simple Lie group with trivial centre and no compact factors, and D istotally disconnected.

Proof. This follows from the Gleason�Montgomery�Zippin solution to Hilbert's �fth prob-lem and the fact that connected semi-simple Lie groups have �nite outer automorphismgroups. More details may be found for example in [Mon01, �11.3]. �

Combining Theorem 4.6 with Theorem 1.6, we �nd the statement given as Corollary 1.7in the Introduction.

Theorem 4.7. Let X 6= R be an irreducible proper CAT(0) space with �nite-dimensionalTits boundary and G < Is(X) any closed subgroup whose action is minimal and does nothave a global �xed point in ∂X.

Then G is either totally disconnected or an almost connected simple Lie group with trivialcentre.

Proof. By Theorem 1.6, G has trivial amenable radical. Let G† be as in Theorem 4.6.Applying Theorem 1.6 to this normal subgroup of G, deduce that we have either G† = Lwith L simple or G† = D. �

We can now complete the proof of Theorem 1.1 and Addendum 1.4 and we adopt theirnotation. Since Is(X) has no global �xed point at in�nity, there is a canonical minimalnon-empty closed convex Is(X)-invariant subset X ′ ⊆ X (Remarks 39 in [Mon06]). Weapply Corollary 4.3 to Z = X ′ and Remark 4.2 to G = Is(Z), setting

G∗ = Is(Rn)× Is(Z1)× · · · × Is(Zm).

All the claimed properties of the resulting factor groups are established in Theorem 1.6,Theorem 1.9 and Theorem 1.19 (the proof of which is completely independent from thepresent considerations). Finally, the claim that any product decomposition of G∗ is aregrouping of the factors in (1.i) is established as follows. Notice that the G∗-action on Z is


still minimal and without �xed point at in�nity (this is almost by de�nition but alternativelyalso follows from Theorem 1.6). Therefore, given any product decomposition of G∗, we canapply the splitting theorem (Corollary 10 in [Mon06]) and obtain a corresponding splittingof Z. Now the uniqueness of the decomposition of the space Z (away from the Euclideanfactor) implies that the given decomposition of G∗ is a regrouping of the factors occurringin Remark 4.2. �

4.C. CAT(0) spaces without Euclidean factor. For the sake of future references, werecord the following consequence of the results obtained thus far:

Corollary 4.8. Let X be a proper CAT(0) space with �nite-dimensional boundary and noEuclidean factor, such that G = Is(X) acts minimally without �xed point at in�nity. ThenG has trivial amenable radical and any subgroup of G acting minimally on X has triv-ial centraliser. Furthermore, given a non-trivial normal subgroup N � G, any N -minimalN -invariant closed subspace of X is a regrouping of factors in the decomposition of Ad-dendum 1.4. In particular, if each irreducible factor of G is non-discrete, then G has nonon-trivial �nitely generated closed normal subgroup.

Proof. The triviality of the amenable radical comes from the corresponding statement inirreducible factors of X, see Theorem 1.6. By the second paragraph of the proof of Theo-rem 1.6, any subgroup of G acting minimally has trivial centraliser. The fact that minimalinvariant subspaces for normal subgroups are �bres in the product decomposition (1.ii) fol-lows since any product decomposition of X is a regrouping of factors in (1.ii) and since anynormal subgroup of G yields such a product decomposition by Theorem 3.3(B.i) and (B.ii).Assume �nally that each irreducible factor in (1.i) is non-discrete and let N < G be a �nitelygenerated closed normal subgroup. Then N is discrete by Baire's category theorem, and Nacts minimally on a �bre, say Y , of the space decomposition (1.ii). Therefore, the projectionof N to Is(Y ) has trivial centraliser, unless N is trivial. Since N is discrete, normal and�nitely generated, its centraliser is open. Since Is(Y ) is non-discrete by assumption, wededuce that N is trivial, as desired. �

5. Totally disconnected group actions

5.A. Smoothness. When considering actions of totally disconnected groups, a desirableproperty is smoothness, namely that points have open stabilisers. This condition is impor-tant in representation theory, but also in our geometric context, see point (ii) of Corollary 5.3below and [Cap07].

In general, this condition does not hold, even for actions that are cocompact, minimaland without �xed point at in�nity. An example will be constructed in Section 11.C below.However, we establish it under a rather common additional hypothesis. Recall that a metricspace X is called geodesically complete (or said to have extensible geodesics) if everygeodesic segment of positive length may be extended to a locally isometric embedding ofthe whole real line. The following contains Theorem 1.15 from the Introduction.

Theorem 5.1. Let G be a totally disconnected locally compact group with a minimal, con-tinuous and proper action by isometries on a proper CAT(0) space X.

If X is geodesically complete, then the action is smooth. In fact, the pointwise stabiliserof every bounded set is open.

Remark 5.2. In particular, the stabiliser of a point acts as a �nite group of isometries onany given ball around this point in the setting of Theorem 5.1.


Corollary 5.3. Let X be a proper CAT(0) space and G be a totally disconnected locallycompact group acting continuously properly on X by isometries. Then:

(i) If the G-action is cocompact, then every element of zero translation length is elliptic.(ii) If the G-action is cocompact and every point x ∈ X has an open stabiliser, then the

G-action is semi-simple.(iii) If the G-action is cocompact and X is geodesically complete, then the G-action is

semi-simple.

Proof of Corollary 5.3. Points (i) and (ii) follow readily from Theorem 5.1, see [Cap07,Corollary 3.3].

(iii) In view of Lemma 2.13, this follows from Theorem 5.1 and (ii). �

The following is a key fact for Theorem 5.1:

Lemma 5.4. Let X be a geodesically complete proper CAT(0) space. Let (Cn)n≥0 be anincreasing sequence of closed convex subsets whose union C =

⋃nCn is dense in X.

Then every bounded subset of X is contained in some Cn; in particular, C = X.

Proof. Suppose for a contradiction that for some r > 0 and x ∈ X the r-ball around x con-tains an element xn not in Cn for each n. We shall construct inductively a sequence {ck}k≥1

of pairwise r-disjoint elements in C with d(x, ck) ≤ 2r + 2, contradicting the properness ofX.

If c1, . . . , ck−1 have been constructed, choose n large enough to that Cn contains themall and d(x,Cn) ≤ 1. Consider the (non-trivial) geodesic segment from xn to its nearestpoint projection xn on Cn; by geodesic completeness, it is contained in a geodesic line andwe choose y at distance r + 1 from Cn on this line. Notice that xn ∈ [xn, y] and henced(y, x) ≤ 2r + 1. Moreover, d(y, ci) ≥ r + 1 for all i < k. Since C is dense, we can chooseck close enough to y to ensure d(ck, x) ≤ 2r + 2 and d(ck, ci) ≥ r for all i < k, completingthe induction step. �

End of proof of Theorem 5.1. The subset C ⊆ X consisting of those points x ∈ X suchthat the stabiliser Gx is open is clearly convex and G-stable. By [Bou71, III �4 No 6], thegroup G contains a compact open subgroup and hence C is non-empty. Thus C is denseby minimality of the action. Since Is(X) is second countable, we can choose a descendingchain Qn < G of compact open subgroups whose intersection acts trivially on X. Therefore,C may be written as the union of an ascending family of closed convex subsets Cn ⊆ X,where Cn is the �xed point set of Qn. Now the statement of the theorem follows fromLemma 5.4. �

5.B. Locally �nite equivariant partitions and cellular decompositions. Let X be alocally �nite cell complex and G be its group of cellular automorphisms, endowed with thetopology of pointwise convergence on bounded subsets. Then G is a totally disconnectedlocally compact group and every bounded subset of X has an open pointwise stabiliser inG. One of the interest of Theorem 5.1 is that it allows for a partial converse to the latterstatement:

Proposition 5.5. Let X be a proper CAT(0) space and G be a totally disconnected locallycompact group acting continuously properly on X by isometries. Assume that the pointwisestabiliser of every bounded subset of X is open in G. Then we have the following:

(i) X admits a canonical locally �nite G-equivariant partition.


(ii) Denoting by σ(x) the piece supporting the point x ∈ X in that partition, we haveStabG(σ(x)) = NG(Gx) and NG(Gx)/Gx acts freely on σ(x).

(iii) If G\X is compact, then so is StabG(σ(x))\σ(x) for all x ∈ X.

Proof. Consider the equivalence relation on X de�ned by

x ∼ y ⇔ Gx = Gy.

This yields a canonical G-invariant partition of X. We need to show that it is locally �nite.Assume for a contradiction that there exists a converging sequence {xn}n≥0 such that thesubgroups Gxn are pairwise distinct. Let x = limn xn.

We claim that Gxn < Gx for all su�ciently large n. Indeed, upon extracting there wouldotherwise exist a sequence gn ∈ Gxn such that gn.x 6= x for all n. Upon a further extraction,we may assume that gn converges to some g ∈ G. By construction g �xes x. Since Gx isopen by hypothesis, this implies that gn �xes x for su�ciently large n, a contradiction. Thisproves the claim.

By hypothesis the pointwise stabiliser of any ball centred at x is open. Thus Gx possessesa compact open subgroup U which �xes every xn. This implies that we have the inclusionU < Gxn < Gx for all n. Since the index of U in Gx is �nite, there are only �nitely manysubgroups of Gx containing U . This �nal contradiction �nishes the proof of (i).

(ii) Straightforward in view of the de�nitions.

(iii) Suppose for a contradiction that H\σ(x) is not compact, where H = StabG(σ(x)). Letthen yn ∈ σ(x) be a sequence such that d(yn, H.x) > n. Let now gn ∈ G be such that{gn.yn} is bounded, say of diameter C. By (i), the set {gnGyng−1

n } is thus �nite. Uponextracting, we shall assume that it is constant. Now, for all n < k, the element g−1

n gknormalises Gyk = Gx and maps yk to a point at distance ≤ C from yn. In view of (ii), thisis absurd. �

Remark 5.6. The partition of X constructed above is non-trivial whenever G does not actfreely. This is for example the case whenever G is non-discrete and acts faithfully.

The pieces in the above partition are generally neither bounded (even if G\X is compact),nor convex, nor even connected. However, if one assumes that the space admits a su�cientlylarge amount of symmetry, then one obtains a partition which deserves to be viewed as anequivariant cellular decomposition.

Corollary 5.7. Let X be a proper CAT(0) space and G be a totally disconnected locallycompact group acting continuously properly on X by isometries. Assume that the pointwisestabiliser of every bounded subset of X is open in G, and that no open subgroup of G�xes a point at in�nity. Then X admits admits a canonical locally �nite G-equivariantdecomposition into compact convex pieces.

Proof. For each x ∈ X, let τ(x) be the �xed-point-set of Gx. Then τ(x) is clearly convex; itis compact by hypothesis. Furthermore the map x 7→ τ(x) is G-equivariant. The fact thatthe collection {τ(x) | x ∈ X} is locally �nite follows from Proposition 5.5. �

5.C. Alexandrov angle rigidity. A further consequence of Theorem 5.1 is a phenomenonof angle rigidity. Given an elliptic isometry g of complete a CAT(0) space X and a pointx ∈ X, we denote by cg,x the projection of x on the closed convex set of g-�xed points.

Proposition 5.8. Let G be a totally disconnected locally compact group with a continuousand proper cocompact action by isometries on a geodesically complete proper CAT(0) spaceX. Then there is ε > 0 such that for any elliptic g ∈ G and any x ∈ X with gx 6= x wehave ∠cg,x(gx, x) ≥ ε.


(We will later also prove an angle rigidity for the Tits angle, see Proposition 6.15.)

Proof. First we observe that this bound on the Alexandrov angle is really a local propertyat cg,x of the germ of the geodesic [cg,x, x] since for any y ∈ [cg,x, x] we have cg,y = cg,x.

Next, we claim that for any n ∈ N, any isometry of order ≤ n of any complete CAT(0)space B satis�es ∠cg,x(gx, x) ≥ 1/n for all x ∈ B that are not g-�xed. Indeed, it followsfrom the de�nition of Alexandrov angles (see [BH99, II.3.1]) that for any y ∈ [cg,x, x] wehave

d(gy, y) ≤ d(cg,x, y)∠cg,x(gx, x).

Therefore, if ∠cg,x(gx, x) < 1/n, the entire g-orbit of y would be contained in a ball aroundy not containing cg,x = cg,y. This is absurd since the circumcentre of this orbit is a g-�xedpoint.

In order to prove the proposition, we now suppose for a contradiction that there aresequences {gn} of elliptic elements in G and {xn} in X with gnxn 6= xn and ∠cn(gnxn, xn)→0, where cn = cgn,xn . Since the G-action is cocompact, there is (upon extracting) a sequence{hn} in G such that hncn converges to some c ∈ X. Upon conjugating gn by hn, replacingxn by hnxn and cn by hncn, we can assume cn → c without loosing any of the conditionson gx, xn and cn, including the relation cn = cgn,xn .

Since d(gnc, c) ≤ 2d(cn, c), we can further extract and assume that {gn} converges to somelimit g ∈ G; notice also that g �xes c. By Lemma 2.13, the action is minimal and henceTheorem 5.1 applies. Therefore, we can assume that all gn coincide with g on some ball Baround c and in particular preserve B. Using Remark 5.2, this provides a contradiction. �

A �rst consequence is an analogue of a result that E. Swenson proved for discrete groups(Theorem 11 in [Swe99]).

Corollary 5.9. Let G be a totally disconnected locally compact group with a continuous andproper cocompact action by isometries on a geodesically complete proper CAT(0) space Xnot reduced to a point.

Then G contains hyperbolic elements (thus in particular elements of in�nite order).

Beyond the totally disconnected case, we can appeal to Theorem 1.1 and Addendum 1.4and state the following.

Corollary 5.10. Let G be any locally compact group with a continuous and proper cocompactaction by isometries on a geodesically complete proper CAT(0) space X not reduced to apoint.

Then G contains elements of in�nite order; if moreover (∂X)G = ∅, then G containshyperbolic elements.

Proof of Corollary 5.9. Proposition 5.8 allows us use the argument form [Swe99]: We canchoose a geodesic ray r : R+ → X, an increasing sequence {ti} going to in�nity in R+ and{gi} in G such that the function t 7→ gir(t + ti) converges uniformly on bounded intervals(to a geodesic line). For i < j large enough, the angle ∠h(r(ti))(r(ti), h

2(r(ti))) de�ned withh = g−1

i gj is arbitrarily close to π. In order to prove that h is hyperbolic, it su�ces to showthat this angle will eventually equal π. Suppose this does not happen; by Corollary 5.3(iii),we can assume that h is elliptic. We set x = r(ti) and c = ch,x. Considering the congruenttriangles (c, x, hx) and (c, hx, h2x), we �nd that ∠c(x, hx) is arbitrarily small. This is incontradiction with Proposition 5.8. �


Proof of Corollary 5.10. If the connected component G◦ is non-trivial, then it contains ele-ments of in�nite order; if it is trivial, we can apply Corollary 5.9.

Assume now (∂X)G = ∅. Then Theorem 1.1 and Addendum 1.4 apply. Therefore, weobtain hyperbolic elements either from Corollary 5.9 or from the fact that any non-compactsemi-simple group contains elements that are algebraically hyperbolic, combined with thefact that the latter act as hyperbolic isometries. That fact is established in Theorem 6.4(i)below, the proof of which is independent of Corollary 5.10. �

5.D. On minimal normal subgroups of t.d. locally compact groups. We begin witha general fact (compare Lemma 1.4.1 in [BM00a]):

Proposition 5.11. Let G be a compactly generated totally disconnected locally compactgroup without non-trivial compact normal subgroups. Then any �ltering family of non-discrete closed normal subgroups has non-trivial (thus non-compact) intersection.

Proof. Let g be a Schreier graph for G. We recall that it consists in choosing any opencompact subgroup U < G (which exists by [Bou71, III �4 No 6]), de�ning the vertex setof g as G/U and drawing edges according to a compact generating set which is a union ofdouble cosets modulo U ; see [Mon01, �11.3]. Since G has no non-trivial compact normalsubgroup, the continuous G-action on g is faithful. Let v0 be a vertex of g and denoteby v⊥0 the set of neighbouring vertices. Since G is vertex-transitive on g, it follows thatfor any normal subgroup N � G, the Nv0-action on v⊥0 de�nes a �nite permutation groupFN < Sym(v⊥0 ) which, as an abstract permutation group, is independent of the choice ofv0. Therefore, if N is non-discrete, this permutation group FN has to be non-trivial sinceU is open and g connected. Now a �ltering family F of non-discrete normal subgroupsyields a �ltering family of non-trivial �nite subgroups of Sym(v⊥0 ). Thus the intersectionof these �nite groups is non-trivial. Let g be a non-trivial element in this intersection. Forany N ∈ F , let Ng be the inverse image of {g} in Nv0 . Thus Ng is a non-empty compactsubset of N for each N ∈ F . Since the family F is �ltering, so are {Nv0 | N ∈ F} and{Ng | N ∈ F}. The result follows, since a �ltering family of non-empty closed subsets ofthe compact set Gv0 has a non-empty intersection. �

Evidently open normal subgroups form a �ltering family; we can thus deduce:

Corollary 5.12. Let G be a compactly generated locally compact group without any non-trivial compact normal subgroup. If G is residually discrete, then it is discrete. �

For convenience, we shall say that a topological group is monolithic with monolith Lif the intersection L of all non-trivial closed normal subgroups is itself non-trivial. A groupis called quasi-simple if it possesses a cocompact normal subgroup which is topologicallysimple and contained in every non-trivial closed normal subgroup of G; in other words, aquasi-simple group is a monolithic group whose monolith is cocompact and topologically sim-ple. The following statement (containing Proposition 1.28 from the Introduction) clari�esthe relation between these notions and a very natural weaker condition. The correspondingstatement in the discrete case is due to J. Wilson [Wil71].

Proposition 5.13. Let G be a compactly generated non-compact locally compact group suchthat every non-trivial closed normal subgroup is cocompact. Then one of the following holds.

(i) G is monolithic; its monolith is the direct product of �nitely many isomorphic topo-logically simple groups.

(ii) G is monolithic with monolith L ∼= Rn. Moreover G/L is isomorphic to a compactirreducible subgroup of GLn(R). In particular G is an almost connected Lie group.


(iii) G is discrete and residually �nite.

Notice that a compactly generated locally compact group with every proper quotientcompact may be monolithic without being quasi-simple. Examples of this nature are indeedprovided by the standard wreath product of a topologically simple group by a �nite transitivepermutation group (see [Wil71, Construction 1]).

Before undertaking the proof of the proposition, we recall that the quasi-centre of alocally compact group G is the subset QZ (G) consisting of all those elements possessingan open centraliser. Clearly QZ (G) is a (topologically) characteristic subgroup of G. Sinceany element with a discrete conjugacy class possesses an open centraliser, it follows that thequasi-centre contains all discrete normal subgroups of G.

Another relevant notion is the following. We say that a subgroup H of a topologicalgroup G is (topologically) locally �nite if every �nite subset of H is contained in acompact subgroup of G. Any locally compact group possesses a maximal normal locally�nite subgroup which is closed and called the LF-radical; another important fact is thatany compact subset of a locally compact topologically locally �nite group is contained in acompact subgroup. We refer to [Pla65] and [Cap07, �2] for details.

It is well known that the LF-radical is compact for connected groups:

Lemma 5.14. Every connected locally compact group admits a maximal compact normalsubgroup. Moreover, the corresponding quotient is a connected Lie group.

Proof. The solution to Hilbert's �fth problem [MZ55, Theorem 4.6] provides a compactnormal subgroup such that the quotient is a Lie group; now the statement follows from thecorresponding fact for connected Lie groups. �

Proof of Proposition 5.13. Since G is non-compact, our assumption implies that it has nonon-trivial compact normal subgroup. We start with a preliminary observation: For anyclosed normal subgroup H � G, the kernel of the representation G → Out(H) is a (notnecessarily direct) product H · ZG(H). Thus, according to whether the normal subgroupZG(H) � G is trivial or cocompact, we see that either G/H injects into Out(H) or theAbelian group Z (H) is cocompact in G, being an intersection of two cocompact normalsubgroups.

We now begin by treating the case where G is totally disconnected. Let L be the inter-section of all non-trivial closed normal subgroups. We distinguish two cases.

If L is trivial, then Proposition 5.11 shows that G admits a non-trivial discrete normalsubgroup Γ; notice that Γ is in�nite. Now Γ is a cocompact lattice in G and is thus �nitelygenerated [Mar91, I.0.40]. In particular Aut(Γ) is countable and thus the image of thecompact group G/Γ in Out(Γ) is �nite since G acts continuously on Γ. By the preliminaryobservation, we deduce that either G/Γ is �nite, in which case (iii) holds, or Γ is virtuallyAbelian. In the latter case, the Abelian group Z Γ is cocompact in G, and G has �niteimage in Out(Z Γ) which now is Aut(Z Γ). Thus G virtually centralises Z Γ. In otherwords G is discrete and virtually Abelian and we are again in case (iii).

Assume now that L is not trivial. Then it is cocompact whence compactly generatedsince G is so. Notice that by de�nition L is characteristically simple. Furthermore L has nonon-trivial compact normal subgroup. Indeed, otherwise L would possess a non-trivial LF-radical and hence L would be topologically locally �nite. Since L is compactly generated,this would imply by [Cap07, Lemma 2.3] that L is compact, which is absurd.

We distinguish two cases.


On the one hand, assume that the quasi-centre QZ (L) is non-trivial. Then it is dense inL. Since L is compactly generated, the arguments of the proof of Theorem 4.8 in [BEW08]show that L possesses a compact open normal subgroup. For the sake of completeness,we brie�y describe them here. Since L is compactly generated, it admits a generating setconsisting of a �nite set {g1, . . . , gn} of elements together with a compact open subgroupU < L. Since L = QZ (L) · U by density of QZ (L), we may assume that each gi belongsto QZ (L). The subgroup

⋂ni=1 ZU (gi) < U is open and hence contains a �nite index open

subgroup V which is normalised by U . Thus V is a compact open normal subgroup of L.Now, since L has no non-trivial compact normal subgroup, we deduce that L is discrete.

The preliminary observation shows that L is of �nite index in G, whence G is discrete aswell. In this case, we are done by the result of J. Wilson [Wil71].

On the other hand, assume that the quasi-centre QZ (L) is trivial. In particular Lpossesses no non-trivial discrete normal subgroup. Proposition 5.11 combined with Zorn'slemma then shows that the set M of non-trivial minimal closed normal subgroups of Lis non-empty. Since L has no non-trivial compact normal subgroup, no element of M iscompact. Some of the following arguments are inspired by the proof of Proposition 1.5.1in [BM00a].

For each subset E ⊆M we setME = 〈M |M ∈ E 〉. We claim that if E is a proper subsetof M , then ME is a proper subgroup of L. Indeed, for all M ∈ E and M ′ ∈M \E we have[M,M ′] ⊆M ∩M ′ = 1. Thus M ′ centralises ME = L. In particular, if M \E is non-emptythen L has a non-trivial centre, which is absurd since QZ (L) = 1 by assumption. Thus theclaim holds indeed.

Clearly G acts on M by conjugation. Let E denote a G-orbit in M . Since ME is normalin G, it is dense in L. The preceding claim thus shows that E = M . In other words G actstransitively on M .

Consider the familyF = {MM \F | F ⊆M is �nite}

of closed normal subgroups of L. Clearly this family is �ltering. Furthermore the aboveclaim shows that MM \F is properly contained in L whenever F is non-empty. Thus

⋂F is

a normal subgroup of G which is properly contained in L. In particular⋂

F is trivial andProposition 5.11 therefore implies that M is �nite.

It now follows that L ∼=∏M∈M M . In particular, any normal subgroup of M is in

fact normalised by L. Since M is a minimal normal subgroup of L, it follows that M istopologically simple and (i) holds.

Now we turn to the case where G is not totally disconnected, hence its identity componentG◦ is cocompact. Since the compact group of Lemma 5.14 is characteristic, it is trivial andhence G◦ is a connected Lie group. Since its soluble radical is characteristic, it is trivial orcocompact.

In the �rst case, G◦ is semi-simple without compact factors. Since its isotypic factors arecharacteristic, there is only one isotypic factor and we conclude that (i) holds.

If on the other hand the radical of G◦ is cocompact, we deduce that G admits a normalcocompact connected soluble subgroup R � G. Since the commutator subgroup of R ischaracteristic and since R has no non-trivial compact subgroup, we have R ∼= Rn. Thekernel of the homomorphism G → Out(R) = Aut(R) is a cocompact normal subgroup Ncontaining containing R in its centre, and such that N/R is compact. In particular N isa compactly generated locally compact group in which every conjugacy class is relativelycompact. By [U²a63], it follows that N is a compact extension of an Abelian group, whichis the direct product of R with a free Abelian group of �nite rank. Since N has trivial


LF-radical, it follows that N is Abelian. Since R is cocompact in N it �nally follows thatN = R. Using again the map G→ Aut(Rn) ∼= GLn(R), we deduce that G/R is isomorphicto a compact subgroup of GLn(R), which has to be irreducible otherwise R would containeda non-cocompact subgroup normalised by G. �

We record one further consequence of Proposition 5.11, concerning locally compact groupsadmitting various quasi-simple quotients. It will be needed below and presents some inde-pendent interest.

Corollary 5.15. Let G be a compactly generated locally compact group and {Nv | v ∈ Σ}be a collection of pairwise distinct closed normal subgroups of G such that for each v ∈ Σ,the quotient Hv = G/Nv is quasi-simple, non-discrete and non-compact. If

⋂v∈ΣNv = 1

then Σ is �nite and G has a characteristic closed cocompact subgroup which splits as a �nitedirect product of |Σ| topologically simple groups.

Proof. We claim that G has no non-trivial compact normal subgroup. Let indeed Q � Gbe a compact normal subgroup of G. By the assumptions made on Hv, the image of Q inHv = G/Nv is trivial for each v ∈ Σ. Thus Q ⊆

⋂v∈ΣNv and hence Q is trivial.

By assumption each quotient Hv is monolithic with simple monolith, which we denote bySv. We claim that Sv is non-discrete. Indeed, if Sv were discrete, then it would be �nitelygenerated since G is compactly generated (see [Mar91, I.0.40]). In particular Aut(Sv) iscountable. Therefore the map G→ Aut(Sv) is not injective. Its kernel is a normal subgroupof G containing Nv but projecting to a non-trivial normal subgroup of Hv meeting Svtrivially, which is absurd. This shows that Sv is indeed non-discrete for each v ∈ Σ.

Let now Mv < G denote the pre-image of Sv in G; thus Mv is the intersection of allclosed normal subgroups of G containing Nv properly. For v 6= v′ ∈ Σ, the normal subgroupMv ∩Mv′ �G is cocompact. Its projection to G/Nv therefore contains the monolith Sv. Inother words Mv < Mv′ . By symmetry Mv′ < Mv and we deduce that Mv is independent ofv ∈ Σ; we denote by M the common value.

Let us now deal with the special case when G is totally disconnected. For each �nitesubset P ⊆ Σ, set NP =

⋂v∈PNv. An elementary argument (compare the Goursat lemma)

shows thatM/NP∼=∏v∈P Sv since each Sv is simple. In particular, arguing with P∪{v} we

deduce that, for each v 6∈ P, the restriction to NP of the map M → Sv is still surjective. Inparticular NP is uncountable, whence non-discrete. Proposition 5.11 therefore shows that ifΣ is in�nite, then the intersection

⋂v∈ΣNv is non-trivial. This yields the desired conclusion

in the totally disconnected case.

We now turn to the general case and denote by G◦ the identity component of G. Theinitial claim shows that the characteristic subgroup of G◦ of Lemma 5.14 is trivial and henceG◦ is a connected Lie group.

Arguing as in the initial claim, G has no non-trivial soluble normal subgroup, and hencethe radical of G◦ is trivial. Thus G◦ is a connected semi-simple Lie group with trivialcentre and no compact factor. It follows (compare Theorem 4.6) that G has a closedcharacteristic subgroup of �nite index which splits as a direct product of the form G◦ ×D. Projecting each Nv to the group of components G/G◦ we obtain a family of normalsubgroups intersecting trivially, each yielding a quasi-simple quotient. We deduce from whathas already been proved that the totally disconnected group G/G◦ has a cocompact closedcharacteristic subgroup which splits as a �nite direct product of non-discrete topologicallysimple groups. �


5.E. Algebraic structure. Given a topological group G, we de�ne its socle soc(G) as thesubgroup generated by all minimal non-trivial closed normal subgroups of G. Notice that Gmight have no minimal non-trivial closed normal subgroup, in which case its socle is trivial.

Proposition 5.16. Let X be a proper CAT(0) space without Euclidean factor and G <Is(X) be a closed subgroup acting minimally cocompactly without �xed point at in�nity. IfG has trivial quasi-centre, then soc(G∗) is direct product of r non-trivial characteristicallysimple groups, where r is the number of irreducible factors of X and G∗ is the canonical�nite index open normal subgroup acting trivially on the set of factors of X.

Proof. We �rst observe that G∗ has no non-trivial discrete normal subgroup. Indeed, sucha subgroup has �nitely many G-conjugates, which implies that each of its elements hasdiscrete G-conjugacy class and hence belongs to QZ (G), which was assumed trivial.

Let now {Ni} be a chain of non-trivial closed normal subgroups of G∗. If Ni is totallydisconnected for some i, then the intersection

⋂iNi is non-trivial by Proposition 5.11.

Otherwise N◦i is non-trivial and normal in (G∗)◦ for each i, and the intersection⋂iNi

is non-trivial by Theorem 1.1 (since the latter describes in particular the possible normalconnected subgroups of G∗). In all cases, Zorn's lemma implies that the ordered set ofnon-trivial closed normal subgroups of G∗ possesses minimal elements.

Given two minimal closed normal subgroups M,M ′, the intersection M ∩ M ′ is thustrivial and, hence, so is [M,M ′]. Thus minimal closed normal subgroups of G∗ centraliseone another. We deduce from Corollary 4.8 that the number of minimal closed normalsubgroups is at most r.

Consider now an irreducible totally disconnected factor H of G∗. We claim that thecollection of non-trivial closed normal subgroups of H forms a �ltering family. Indeed,given two such normal subgroup N1, N2, then N1 ∩N2 is again a closed normal subgroup ofH. It is is trivial, then the commutator [N1, N2] is trivial and, hence, the centraliser of N1 inH is non-trivial, contradicting Theorem 1.6. This con�rms the claim. Thus the intersectionof all non-trivial closed normal subgroups of H is non-trivial by Proposition 5.11. Clearlythis intersection is the socle of H; it is clear we have just established that it is containedin every non-trivial closed normal subgroup of H. In particular soc(H) is characteristicallysimple. The desired result follows, since soc(H) is clearly a minimal closed normal subgroupof G∗. �

Theorem 5.17. Let X be a proper irreducible geodesically complete CAT(0) space. LetG < Is(X) be a closed totally disconnected subgroup acting cocompactly, in such a way thatno open subgroup �xes a point at in�nity. Then we have the following:

(i) Every compact subgroup of G is contained in a maximal one; the maximal compactsubgroups fall into �nitely many conjugacy classes.

(ii) QZ (G) = 1.(iii) soc(G) is a non-discrete characteristically simple group.

Proof. (i) By Lemma 2.13, the action is minimal and hence Theorem 5.1 applies. In par-ticular, we can apply Corollary 5.7 and consider the resulting equivariant decomposition.Let Q < G be a compact subgroup and x be a Q-�xed point. If Gx is not contained ina maximal compact subgroup of G, then there is an in�nite sequence (xn)n≥0 such thatx0 = x and Gxn ⊆ Gxn+1 . By Corollary 5.7, the sequence xn leaves every bounded subset.Since the �xed points XGxn form a nested sequence, it follows that XGx is unbounded. Inparticular its visual boundary ∂(XGx) is non-empty and the open subgroup Gx has a �xedpoint at in�nity. This contradicts the hypotheses, and the claim is proved. Notice that


a similar argument shows that for each x ∈ X, there are �nitely many maximal compactsubgroups Qi < G containing Gx.

The fact that G possesses �nitely many conjugacy classes of maximal compact subgroupsnow follows from the compactness of G\X.

(ii) We claim that QZ (G) is topologically locally �nite. The desired result follows since itis then amenable but G has trivial amenable radical by Theorem 1.6. Let S ⊆ QZ (G) be a�nite subset. Then G possesses a compact open subgroup U centralising S. By hypothesisthe �xed point set of U is compact. Since 〈S〉 stabilises XU , it follows that 〈S〉 is compact,whence the claim.

(iii) Follows from (ii) and Proposition 5.16. �

6. Cocompact CAT(0) spaces

6.A. Fixed points at in�nity. We begin with a simple observation. We recall that twopoints at in�nity are opposite if they are the two endpoints of a geodesic line. We denoteby ξop the set of points opposite to ξ. Recall from [Bal95, Theorem 4.11(i)] that, if X isproper, then two points ξ, η ∈ ∂X at Tits distance > π are necessarily opposite. (Recallthat Tits distance is by de�nition the length metric associated to the Tits angle.) However,it is not true in general that two points at Tits distance π are opposite.

Proposition 6.1. Let X be a proper CAT(0) space and H < Is(X) a closed subgroup actingcocompactly. If H �xes a point ξ at in�nity, then ξop 6= ∅ and H acts transitively on ξop.

Proof. First we claim that there is a geodesic line σ : R → X with σ(∞) = ξ. Indeed, letr : R+ → X ′ be a ray pointing to ξ and {gn} a sequence in H such that gnr(n) remainsbounded. The Arzelà�Ascoli theorem implies that gnr(R+) subconverges to a geodesic linein X. Since ξ is �xed by all gn, this line has an endpoint at ξ.

Let now σ′ : R → X be any other geodesic with σ′(∞) = ξ and choose a sequence{hn}n∈N in H such that d(hnσ(−n), σ′(−n)) remains bounded. By convexity and sinceall hn �x ξ, d(hnσ(t), σ′(t)) is bounded for all t and thus subconverges (uniformly for t inbounded intervals). On the one hand, it implies that {hn} has an accumulation point h.On the other hand, it follows that hσ(−∞) = σ′(−∞). �

Recall that any complete CAT(0) space X admits a canonical splitting X = X ′ × Vpreserved by all isometries, where V is a (maximal) Hilbert space called the Euclideanfactor of X, see [BH99, II.6.15(6)]. Furthermore, there is a canonical embedding X ′ ⊆X ′′×V ′, where V ′ is a Hilbert space generated by all directions inX ′ pointing to ��at points�at in�nity, namely points for which the Busemann functions are a�ne on X ′; moreover,every isometry of X ′ extends uniquely to an isometry of X ′′ × V ′ which preserves thatsplitting. This is a result of Adams�Ballmann [AB98a, Theorem 1.6], who call V ′ thepseudo-Euclidean factor (one could also propose �Euclidean pseudo-factor�).

Corollary 6.2. Let X be a proper CAT(0) space with a cocompact group of isometries.Then the pseudo-Euclidean factor of X is trivial.

Proof. In view of the above discussion, X ′ is also a proper CAT(0) space with a cocompactgroup of isometries. The set of �at points in ∂X ′ admits a canonical (intrinsic) circumcentreξ by Lemma 1.7 in [AB98a]. In particular, ξ is �xed by all isometries and therefore, byProposition 6.1, it has an opposite point, which is impossible for a �at point unless it liesalready in the Euclidean factor (see [AB98a]). �


Proposition 6.3. Let G be a group acting cocompactly by isometries on a proper CAT(0)space X without Euclidean factor and assume that the stabiliser of every point at in�nityacts minimally on X. Then G has no �xed point at in�nity.

Proof. If G has a global �xed point ξ, then the stabiliser Gη of an opposite point η ∈ ξop

(which exists by Proposition 6.1) preserves the union Y ⊆ X of all geodesic lines connectingξ to η. By [BH99, II.2.14], this space is convex and splits as Y = R × Y0. Since Gη actsminimally, we deduce Y = X which provides a Euclidean factor. �

6.B. Actions of simple algebraic groups. Let X be a CAT(0) space and G be an alge-braic group de�ned over the �eld k. An isometric action of G(k) on X is called algebraicif every (algebraically) semi-simple element g ∈ G(k) acts as a semi-simple isometry.

Theorem 6.4. Let k be a local �eld and G be an absolutely almost simple simply connectedk-group. Let X be a non-compact proper CAT(0) space on which G = G(k) acts continuouslyby isometries.

Assume either: (a) the action is cocompact; or: (b) it has full limit set, is minimal and∂X is �nite-dimensional. Then:

(i) The G-action is algebraic.(ii) There is a G-equivariant bijection ∂X ∼= ∂Xmodel which is a homeomorphism with

respect to the cône topology and an isometry with respect to Tits' metric, whereXmodel is the symmetric space or Bruhat�Tits building associated with G.

(iii) If X is geodesically complete, then X is isometric to Xmodel.(iv) For any semi-simple k-subgroup L < G, there non-empty closed convex subspace

Y ⊆ X minimal for L = L(k); moreover, there is no L-�xed point in ∂Y .

Remarks 6.5.

(i) Notice that there is no assumption on the k-rank of G in this result.(ii) We recall for (b) that minimality follows from full limit set in the geodesically

complete case (Lemma 2.13).(iii) Recall that two CAT(0) spaces with the same cocompact isometry group need not

have homeomorphic boundaries [CK00].

Before proceeding to the proof, we give two examples showing that the assumptions madein Theorem 6.4 are necessary.

Example 6.6. Without the assumption of geodesic completeness, it is not true in general that,in the setting of the theorem, the space X contains a closed convex G-invariant subspacewhich is isometric to Xmodel. A simple example of this situation may obtained as follows.Consider the case where k is non-Archimedean and G has k-rank one. Let 0 < r < 1/2 andlet X be the space obtained by replacing the r-ball centred at each vertex in the tree Xmodel

by an isometric copy of a given Euclidean n-simplex, where n + 1 is valence of the vertex.In this way, one obtains a CAT(0) space which is still endowed with an isometric G-actionthat is cocompact and minimal, but clearly X is not isometric to Xmodel.

We do not know whether such a construction may also be performed in the Archimedeancase (see Problem 12.2 below).

Example 6.7. Under the assumptions (b), minimality is needed. Indeed, we claim that forany CAT(0) space X0 there is a canonical CAT(−1) space X (in particular X is a CAT(0)space) together with a canonical map i : Is(X0) ↪→ Is(X) with the following properties: Theboundary ∂X is reduced to a single point; X non-compact; X is proper if and only if X0 isso; the map i is an isomorphism of topological groups onto its image. This claim justi�es


that minimality is needed since we can apply it to the case where X0 is the symmetric spaceor Bruhat�Tits building associated to G(k). (In that case the action has indeed full limit set,a cheap feat as the isometry group is non-compact and the boundary rather incapacious.)

To prove the claim, consider the parabolic cône Y associated to X0. This is the metricspace with underlying set X0×R∗+ where the distance is de�ned as follows: given two points(x, t) and (x′, t′) of Y , identify the interval [x, x′] ⊆ X0 with an interval of correspondinglength in R and measure the length from the resulting points (x, t) and (x′, t′) in the upperhalf-plane model for the hyperbolic plane. This is a particular case of the synthetic version([Che99], [AB98b]) of the Bishop�O'Neill �warped products� [BO69] and its properties aredescribed in [BGP92], [AB04, 1.2(2A)] and [HLS00, �2]. In particular, Y is CAT(−1).

We now let ξ ∈ ∂Y be the point at in�nity corresponding to t → ∞ and de�ne X ⊆ Yto be an associated horoball; for de�niteness, set X = X0× [1,∞). We now have ∂X = {ξ}by the CAT(−1) property or alternatively by the explicit description of geodesic rays (e.g.2(iv) in [HLS00]). The remaining properties follow readily.

Proof of Theorem 6.4. We start with a few preliminary observations. Finite-dimensionalityof the boundary always holds since it is automatic in the cocompact case. Since X is non-compact, the action is non-trivial, because it has full limit set. It is well known that everynon-trivial continuous homomorphism of G to a locally compact second countable group isproper [BM96, Lemma 5.3]. Thus the G-action on X is proper.

We claim that the stabiliser of any point ξ ∈ ∂X contains the unipotent radical of someproper parabolic subgroup of G. Indeed, �x a polar decomposition G = KAK. Let x0 ∈ Xbe a K-�xed point. Choose a sequence {gn}n≥0 of elements of G such that gn.x0 convergesto ξ. Write gn = kn.an.k

′n with kn, k

′n ∈ K and an ∈ A. We may furthermore assume,

upon replacing {gn} by a subsequence, that {kn} converges to some k ∈ K, that {an.x0}converges in X ∪ ∂X and that {an.p} converges in Xmodel ∪ ∂Xmodel, where p ∈ Xmodel issome base point. Let η = limn→∞ an.x0 and observe η = k−1ξ. Furthermore, the stabiliserof η contains the group

U = {g ∈ G | limn→∞

a−1n gan = 1}.

The convergence in direction of {an} in A implies that U contains the unipotent radicalUQ of the parabolic subgroup Q < G corresponding to limn→∞ an.p ∈ ∂Xmodel. (In fact,the arguments for Lemma 2.4 in [Pra77] probably show U = UQ; this follows a posteriorifrom (ii) below.) Therefore, the stabiliser of ξ = k.η in G contains the unipotent radical ofkQk−1, proving the claim.

Notice that we have seen in passing that any point at in�nity lies in the limit set of sometorus; in the above notation, ξ is in the limit set of kAk−1.

(i) Every element of G which is algebraically elliptic acts with a �xed point in X, since itgenerates a relatively compact subgroup. We need to show that every non-trivial element ofa maximal split torus T < G acts as a semi-simple isometry. Assume for a contradiction thatsome element t ∈ T acts as a parabolic isometry. Since X has �nite-dimensional boundaryand we can apply Corollary 2.3(ii). It follows that the Abelian group T has a canonical�xed point at in�nity ξ �xed by the normaliser NG(T ). By the preceding paragraph, weknow furthermore that the stabiliser of ξ in G also contains the unipotent radical of someparabolic subgroup of G. Recall that G is generated by NG(T ) together with any suchunipotent radical: this follows from the fact that NG(T ) has no �xed point at in�nity inXmodel and that G is generated by the unipotent radicals of any two distinct parabolicsubgroups. Therefore ξ is �xed by the entire group G. Since G has trivial Abelianisation,its image under the Busemann character centred at ξ vanishes, thereby showing that G must


stabilise every horoball centred at ξ. This is absurd both in the minimal and the cocompactcase.

(ii) Let T < G be a maximal split torus. Let Fmodel ⊆ Xmodel be the (maximal) �atstabilised by T . In view of (i) and the properness of the T -action, we know that T alsostabilises a �at F ⊆ X with dimF = dimT , see [BH99, II.7.1]. Choose a base pointp0 ∈ Fmodel in such a way that its stabiliser K := Gp0 is a maximal compact subgroup ofG. The union of all T -invariant �ats which are parallel to F is NG(T )-invariant. Therefore,upon replacing F by a parallel �at, we may � and shall � assume that F contains a pointx0 which is stabilised by NK := NG(T ) ∩K. Note that, since NG(T ) = 〈NK ∪ T 〉, the �atF is NG(T )-invariant. Therefore, there is a well de�ned NG(T )-equivariant map α of theNG(T )-orbit of p0 to F , de�ned by α(g.p0) = g.x0 for all g ∈ NG(T ).

We claim that, up to a scaling factor, the map α is isometric. In order to establishthis, remark that the Weyl group W := NG(T )/ZG(T ) acts on F , since W = NK/TK ,where TK := ZG(T ) ∩K acts trivially on F . The group NK normalises the coroot latticeΛ < T . Furthermore NK .Λ acts on Fmodel as an a�ne Weyl group since NK .Λ/TK ∼= WnΛ.Moreover, since any re�ection in W centralises an Abelian subgroup of corank 1 in Λ, itfollows that NK .Λ acts on F as a discrete re�ection group. But a given a�ne Weyl group hasa unique (up to scaling factor) discrete cocompact action as a re�ection group on Euclideanspaces, as follows from [Bou68, Ch. VI, �2, Proposition 8]. Therefore the restriction of α toΛ.x0 is a homothety. Since Λ is a uniform lattice in T , the claim follows.

At this point, it follows that α induces an NG(T )-equivariant map ∂α : ∂Fmodel → ∂F ,which is isometric with respect to Tits' distance. Since NG(T ) is the stabiliser of ∂Fmodel inG, this map extends to a well de�ned G-equivariant map ∂Xmodel → ∂X, which we denoteagain by ∂α. Since any two points of ∂Xmodel are contained in a common maximal sphere(i.e. an apartment), and since G acts transitively on these spheres, the map ∂α is isometric,because so is its restriction to the sphere ∂Fmodel. Note that ∂α is surjective: indeed, thisfollows from the last preliminary observation, which, combined with (i), shows in particularthat ∂X = K.∂F .

It remains to prove that ∂α is a homeomorphism with respect to the cône topology.Since ∂Xmodel is compact, it is enough to show that ∂α is continuous. Now any convergentsequence in ∂Xmodel may be written as {kn.ξn}n≥0, where {kn}n≥0 (resp. {ξn}n≥0) is aconvergent sequence of elements of K (resp. ∂Fmodel). On the sphere ∂Fmodel, the cônetopology coincides with the one induced by Tits' metric. Therefore, the equivariance of theTits' isometry ∂α shows that {∂α(kn.ξn)}n≥0 is a convergent sequence in ∂X, as was to beproved.

(iii) In the higher rank case, assertion (iii) follows from (ii) and the main result of [Lee00].However, the full strength of loc. cit. is really not needed here, since the main di�cultythere is precisely the absence of any group action, which is part of the hypotheses in oursetting. For example, when the ground �eld k is the �eld of real numbers, the argumentsmay be dramatically shortened as follows; they are valid without any rank assumption.

Given any ξ ∈ ∂X, the unipotent radical of the parabolic subgroup Gξ acts sharplytransitively on the boundary points opposite to ξ. In view of this and of the properness ofthe G-action, the arguments of [Lee00, Proposition 4.27] show that geodesic lines in X donot branch; in other words X has uniquely extensible geodesics. From this, it follows thatthe group NK = NG(T ) ∩K considered in the proof of (ii) has a unique �xed point in X,since otherwise it would �x pointwise a geodesic line, and hence, by (ii), opposite points in


∂Xmodel. The fact that this is impossible is purely a statement on the classical symmetricspace Xmodel; we give a proof for the reader's convenience:

Let Fmodel be the �at corresponding to T and p0 ∈ Fmodel be the K-�xed point. IfNK �xed a point ξ ∈ ∂Xmodel, then the ray [p0, ξ) would be pointwise �xed and, hence,the group NK would �x a non-zero vector in the tangent space of Xmodel at p0. A Cartandecomposition g = k⊕p of the Lie algebra g of G yields an isomorphism between the isotropyrepresentation of NK on Tp0Xmodel and the representation of the Weyl group W on p. Aneasy explicit computation shows that the latter representation has no non-zero �xed vector.

Since NK has a unique �xed point, the latter is stabilised by the entire group K. HenceK �xes a point lying on a �at F stabilised by T . From the KAK-decomposition, it followsthat the G-orbit of this �xed point is convex. Since the G-action on X is minimal bygeodesic completeness (Lemma 2.13), we deduce that G is transitive onX. In particularX iscovered by �ats which are G-conjugate to F , and the existence of a G-equivariant homothetyXmodel → X follows from the existence of a NG(T )-equivariant homothety Fmodel → F ,which has been established above. It remains only to choose the right scale on Xmodel tomake it an isometry.

In the non-Archimedean case, we consider only the rank one case, referring to [Lee00] forhigher rank. Let K be a maximal compact subgroup of G and x0 ∈ X be a K-�xed point.By (ii), the group K acts transitively on ∂X. Since X is geodesically complete, it followsthat the K-translates of any ray emanating from x0 cover X entirely. On the other handevery point in X has an open stabiliser by Theorem 5.1, any point in X has a �nite K-orbit.This implies that the space of directions at each point p ∈ X is �nite. In other words X is1-dimensional. Since X is CAT(0) and locally compact, it follows that X is a locally �nitemetric tree. As we have just seen, the group K acts transitively on the geodesic segmentsof a given length emanating from x0. One deduces that G is transitive on the edges of X.In particular all edges of X have the same length, which we can assume to be as in Xmodel,G has at most two orbits of vertices, and X is either regular or bi-regular. The valence ofany vertex p equals

ming 6∈NG(Gp)

[Gp : Gp ∩ gGpg−1

],

and coincides therefore with the valence of Xmodel. It �nally follows that X and Xmodel areisometric, as was to be proved.

(iv) Let P be a k-parabolic subgroup of G that is minimal amongst those containing L.We may assume P 6= G since otherwise L has no �xed point in ∂X and the conclusionholds in view of Proposition 3.1. It follows that L centralises a k-split torus T of positivedimension d. It follows from (i) that T = T(k) acts by hyperbolic isometries, and thusthere is a T -invariant closed convex subset Z ⊆ X of the form Z = Z1 × Rd such thatthe T -action is trivial on the Z1 factor; this follows from Theorem II.6.8 in [BH99] and theproperness of the action. Moreover, L preserves Z and its decomposition Z = Z1 × Rd,acting by translations on the Rd factor (loc. cit.). Since L is semi-simple, this translationaction is trivial and thus L preserves any Z1 �bre, say for instance Z0 := Z1 × {0} ⊆ Z.For both the existence of a minimal set Y and the condition (∂Y )L = ∅, it su�ces to showthat L has no �xed point in ∂Z0 (Proposition 3.1).

We claim that ∂Z0 is Tits-isometric to the spherical building of the Lévi subgroup ZG(T).Indeed, we know from (ii) that ∂X is equivariantly isometric to ∂Xmodel and the building ofZG(T) is characterised as the points at distance π/2 from the boundary of the T -invariant�at in ∂Xmodel.


On the other hand, L has maximal semi-simple rank in ZG(T) by the choice of P andtherefore cannot be contained in a proper parabolic subgroup of ZG(T). This shows thatL has no �xed point in ∂Z0 and completes the proof. �

6.C. No branching geodesics. Recall that in a geodesic metric space X, the space of

directions Σx at a point x is the completion of the space Σx of geodesic germs equipped withthe Alexandrov angle metric at x. If X has uniquely extensible geodesics, then Σx = Σx.

The following is a result of V. Berestovskii [Ber02] (we read it in [Ber, �3]; it also followsfrom A. Lytchak's arguments in [Lyt05, �4]).

Theorem 6.8. Let X be a proper CAT(0) space with uniquely extensible geodesics and

x ∈ X. Then Σx = Σx is isometric to a Euclidean sphere. �

We use this result to establish the following.

Proposition 6.9. Let X be a proper CAT(0) space with uniquely extensible geodesics. Thenany totally disconnected closed subgroup D < Is(X) is discrete.

Proof. There is some compact open subgroup Q < D, see [Bou71, III �4 No 6]. Let x bea Q-�xed point. The isometry group of Σx is a compact Lie group by Theorem 6.8 andthus the image of the pro�nite group Q in it is �nite. Let thus K < Q be the kernelof this representation, which is open. Denote by S(x, r) the r-sphere around x. The Q-equivariant �visual� map S(x, r)→ Σx is a bijection by unique extensibility. It follows thatK is trivial. �

We are now ready for:

End of proof of Theorem 1.11. Since the action is cocompact, it is minimal by Lemma 2.13.The fact that extensibility of geodesics is inherited by direct factors of the space followsfrom the characterisation of geodesics in products, see [BH99, I.5.3(3)]. Each factor Xi isthus a symmetric space in view of Theorem 6.4(iii). By virtue of Corollary 5.3(iii), thetotally disconnected factors Dj act by semi-simple isometries.

Assume now that X has uniquely extensible geodesics. For the same reason as before,this property is inherited by each direct factor of the space. Thus each Dj is discrete byProposition 6.9. �

Theorem 6.10. Let X be a proper irreducible CAT(0) space with uniquely extensible geodesics.If X admits a non-discrete group of isometries with full limit set but no global �xed point atin�nity, then X is a symmetric space.

The condition on �xed points at in�nity is necessary in view of E. Heintze's examples [Hei74]of negatively curved homogeneous manifolds which are not symmetric spaces. In fact thesespaces consist of certain simply connected soluble Lie groups endowed with a left-invariantnegatively curved Riemannian metric.

Proposition 6.11. Let X be a proper CAT(0) space with uniquely extensible geodesics.Then ∂X has �nite dimension.

Proof. Let x ∈ X and recall that by Berestovskii's result quoted in Theorem 6.8 above, Σx isisometric to a Euclidean sphere. By de�nition of the Tits angle, the �visual� map ∂X → Σx

associating to a geodesic ray its germ at x is Tits-continuous (in fact, 1-Lipschitz). It isfurthermore injective (actually, bijective) by unique extensibility. Therefore, the topologicaldimension of any compact subset of ∂X is bounded by the dimension of the sphere Σx. The


claim follows now from Kleiner's characterisation of the dimension of spaces with curva-ture bounded above in terms of the topological dimension of compact subsets (Theorem Ain [Kle99]). �

Proof of Theorem 6.10. By Lemma 2.13, the action of G := Is(X) is minimal. In view ofProposition 6.11, the boundary ∂X is �nite-dimensional. Thus we can apply Theorem 1.1and Addendum 1.4. Since X is irreducible and non-discrete, Proposition 6.9 implies thatG is an almost connected simple Lie group (unless X = R, in which case X is indeed asymmetric space). We conclude by Theorem 6.4. �

6.D. No open stabiliser at in�nity. The following statement sums up some of the pre-ceding considerations:

Corollary 6.12. Let X be a proper geodesically complete CAT(0) space without Euclideanfactor such that some closed subgroup G < Is(X) acts cocompactly. Suppose that no opensubgroup of G �xes a point at in�nity. Then we have the following:

(i) X admits a canonical equivariant splitting

X ∼= X1 × · · · ×Xp × Y1 × · · · × Yqwhere each Xi is a symmetric space and each Yj possesses a G-equivariant locally�nite decomposition into compact convex cells.

(ii) G possesses hyperbolic elements.(iii) Every compact subgroup of G is contained in a maximal one; the maximal compact

subgroups fall into �nitely many conjugacy classes.(iv) QZ (G) = 1; in particular G has no non-trivial discrete normal subgroup.(v) soc(G∗) is a direct product of p+ q non-discrete characteristically simple groups.

Proof. (i) Follows from Theorem 1.11 and Corollary 5.7.

(ii) Clear from Corollary 5.10.

(iii) and (iv) Immediate from (i) and Theorem 5.17(i) and (ii).

(v) Follows from (i), (iv) and Proposition 5.16. �

Proof of Theorems 1.13 and 1.14. It su�ces to observe that the above proof of (i) is immuneto Euclidean factors (as is the statement of (i) itself actually). �

6.E. Cocompact stabilisers at in�nity. We undertake the proof of Theorem 1.18 whichdescribes isometrically any geodesically complete proper CAT(0) space such that the sta-biliser of every point at in�nity acts cocompactly.

Remark 6.13. (i) The formulation of Theorem 1.18 allows for symmetric spaces of Eu-clidean type. (ii) A Bass�Serre tree is a tree admitting an edge-transitive automorphismgroup; in particular, it is regular or bi-regular (the regular case being a special case ofEuclidean buildings).

Lemma 6.14. Let X be a proper CAT(0) space such that the stabiliser of every point atin�nity acts cocompactly on X. For any ξ ∈ ∂X, the set of η ∈ ∂X with ∠T(ξ, η) = π iscontained in a single orbit under Is(X).

Proof. Write G = Is(X). In view of Proposition 6.1 applied to Gξ, it su�ces to prove thatthe G-orbit of any such η contains a point opposite to ξ. By de�nition of the Tits angle,there is a sequence {xn} in X such that ∠xn(ξ, η) tends to π. Since Gξ acts cocompactly,it contains a sequence {gn} such that, upon extracting, gnxn converges to some x ∈ Xand gnη to some η′ ∈ ∂X. The angle semi-continuity arguments given in the proof of


Proposition II.9.5(3) in [BH99] show that ∠x(ξ, η′) = π, recalling that all gn �x ξ. Thismeans that there is a geodesic σ : R→ X through x with σ(−∞) = ξ and σ(∞) = η′. Onthe other hand, since Gη is cocompact in G, the G-orbit of η is closed in the cône topology.This means that there is g ∈ G with η′ = gη, as was to be shown. �

We shall need another form of angle rigidity (compare Proposition 5.8), this time for Titsangles.

Proposition 6.15. Let X be a geodesically complete proper CAT(0) space, G < Is(X) aclosed totally disconnected subgroup and ξ ∈ ∂X. If the stabiliser Gξ acts cocompactly onX, then the G-orbit of ξ is discrete in the Tits topology.

Proof. Suppose for a contradiction that there is a sequence {gn} such that gnξ 6= ξ for all nbut ∠T(gnξ, ξ) tends to zero. Since Gξ is cocompact, we can assume that gn converges in G;since the Tits topology is �ner than the cône topology for which the G-action is continuous,the limit of gn must �x ξ and we can therefore assume gn → 1. Let B ⊆ X be an open balllarge enough so that Gξ.B = X. Since by Lemma 2.13 we can apply Theorem 5.1, there isno loss of generality in assuming that each gn �xes B pointwise.

Let c : R+ → X be a geodesic ray pointing towards ξ with c(0) ∈ B. For each n there isrn > 0 such that c and gnc branch at the point c(rn). In particular, gn �xes c(rn) but notc(rn + ε) no matter how small ε > 0. We now choose hn ∈ Gξ such that xn := hnc(rn) ∈ Band notice that the sequence kn := hngnh

−1n is bounded since kn �xes xn. We can therefore

assume upon extracting that it converges to some k ∈ G; in view of Theorem 5.1, we canfurther assume that all kn coincide with k on B and in particular k �xes all xn. Since∠T(knξ, ξ) = ∠T(gnξ, ξ), we also have k ∈ Gξ. Considering any given n, it follows now thatk �xes the ray from xn to ξ. Thus kn �xes an initial segment of this ray at xn. This isequivalent to gn �xing an initial segment at c(rn) of the ray from c(rn) to ξ, contrary to ourconstruction. �

Here is a �rst indication that our spaces might resemble symmetric spaces or Euclideanbuildings:

Proposition 6.16. Let X be a proper CAT(0) space such that that the stabiliser of everypoint at in�nity acts cocompactly on X. Then any point at in�nity is contained in anisometrically embedded standard n-sphere in ∂X, where n = dim ∂X.

Proof. Let η ∈ ∂X. There is some standard n-sphere S isometrically embedded in ∂Xbecause X is cocompact (Theorem C in [Kle99]). By Lemma 3.1 in [BL05], there is ξ ∈ Swith ∠T(ξ, η) = π. Let ϑ ∈ S be the antipode in S of ξ. In view of Lemma 6.14, there isan isometry sending ϑ to η. The image of S contains η. �

We need one more fact for Theorem 1.18. The boundary of a CAT(0) space need notbe complete, regardless of the geodesic completeness of the space itself; however, this is thecase in our situation in view of Proposition 6.16:

Corollary 6.17. Let X be a proper CAT(0) space such that that the stabiliser of every pointat in�nity acts cocompactly on X. Then ∂X is geodesically complete.

Proof. Suppose for a contradiction that some Tits-geodesic ends at ξ ∈ ∂X and let B ⊆∂X be a small convex Tits-neighbourhood of ξ; in particular, B is contractible. Since byProposition 6.16 there is an n-sphere through ξ for n = dim ∂X, the relative homologyHn(B,B \ {ξ}) is non-trivial. Our assumption implies that B \ {ξ} is contractible by usingthe geodesic contraction to some point η ∈ B \ {ξ} on the given geodesic ending at ξ.


This implies Hn(B,B \ {ξ}) = 0, a contradiction. (This argument is adapted from [BH99,II.5.12].) �

End of proof of Theorem 1.18. We shall use below that product decompositions preservegeodesic completeness (this follows e.g. from [BH99, I.5.3(3)]). We can reduce to the casewhere X has no Euclidean factor. By Lemma 2.13, the group G = Is(X) as well as allstabilisers of points at in�nity act minimally. In particular, Proposition 6.3 ensures that Ghas no �xed point at in�nity and we can apply Theorem 1.1 and Addendum 1.4. Therefore,we can from now on assume that X is irreducible. If the identity component G◦ is non-trivial, then Theorem 1.11 (see also Theorem 6.4(iii)) ensures that X is a symmetric space,and we are done. We assume henceforth that G is totally disconnected.

For any ξ ∈ ∂X, the collection Ant(ξ) = {η : ∠T(ξ, η) = π} of antipodes is contained ina G-orbit by Lemma 6.14 and hence is Tits-discrete by Proposition 6.15. This discretenessand the geodesic completeness of the boundary (Corollary 6.17) are the assumptions neededfor Proposition 4.5 in [Lyt05], which states that ∂X is a building. Since X is irreducible,∂X is not a (non-trivial) spherical join, see Theorem II.9.24 in [BH99]. Thus, if this buildinghas non-zero dimension, we conclude from the main result of [Lee00] that X is a Euclideanbuilding of higher rank.

If on the other hand ∂X is zero-dimensional, then we claim that it is homogeneous underG. Indeed, we know already that for any given ξ ∈ ∂X, the set Ant(ξ) lies in a single orbit.Since in the present case Ant(ξ) is simply ∂X \ {ξ}, the claim follows from the fact that Ghas no �xed point at in�nity. Now we argue as in the proof of Theorem 6.4(iii) and deducethat X is an edge-transitive tree. �


Part II. Discrete subgroups of the isometry group

7. An analogue of Borel density

Before discussing our analogue of Borel's density theorem [Bor60] in Section 7.B below,we present a more elementary phenomenon based on co-amenability.

7.A. Fixed points at in�nity. Recall that a subgroup H of a topological group G isco-amenable if any continuous a�ne G-action on a convex compact set (in a Hausdor�locally convex topological vector space) has a �xed point whenever it has an H-�xed point.The arguments of Adams�Ballmann [AB98a] imply the following preliminary step towardsTheorem 7.4:

Proposition 7.1. Let G be a topological group with a continuous isometric action on aproper CAT(0) space X without Euclidean factor. Assume that the G-action is minimal anddoes not have a global �xed point in ∂X.

Then any co-amenable subgroup of G still has no global �xed point in ∂X.

Proof. Suppose for a contradiction that a co-amenable subgroup H < G �xes ξ ∈ ∂X. ThenG preserves a probability measure µ on ∂X and we obtain a convex function f : X → R byintegrating Busemann functions against this measure; as in [AB98a], the cocycle equation forBusemann functions (see �1.A) imply that f is G-invariant up to constants. The argumentstherein show that f is constant and that µ is supported on �at points. However, in theabsence of a Euclidean factor, the set of �at points has a unique circumcentre when non-empty [AB98a, 1.7]; this provides a G-�xed point, a contradiction. �

Combining the above with the splitting methods used in Theorem 3.3, we record a conse-quence showing that the exact conclusions of the Adams�Ballmann theorem [AB98a] holdunder much weaker assumptions than the amenability of G.

Corollary 7.2. Let G be a topological group with a continuous isometric action on a properCAT(0) space X. Assume that G contains two commuting co-amenable subgroups.

Then either G �xes a point at in�nity or it preserves a Euclidean subspace in X.

We emphasise that one can easily construct a wealth of examples of highly non-amenablegroups satisfying these assumptions. For instance, given any group Q, the restricted wreathproduct G = Z n

⊕n∈ZQ contains the pair of commuting co-amenable groups H+ =⊕

n≥0Q and H− =⊕

n<0Q, see [MP03]. (In fact, one can even arrange for H± to beconjugated upon replacing Z by the in�nite dihedral group.)

For similar reasons, we deduce the following �xed-point property for R. Thompson's group

F :=⟨gi, i ∈ N | g−1

i gjgi = gj+1 ∀ j > i⟩;

this �xed-point result explains why the strategy proposed in [Far08b] to disprove amenabilityof F with the Adams�Ballmann theorem cannot work.

Corollary 7.3. Any F -action by isometries on any proper CAT(0) space X has a �xedpoint in X.

Proof of Corollary 7.2. We assume that G has no �xed point at in�nity. By Proposition 3.1,there is a minimal non-empty closed convex G-invariant subspace. Upon considering theEuclidean decomposition [BH99, II.6.15] of the latter, we can assume that X is G-minimaland without Euclidean factor and need to show that G �xes a point in X.

Let H± < G be the commuting co-amenable groups. In view of Proposition 7.1, bothact without �xed point at in�nity. In particular, we have an action of H = H+ × H−


without �xed point at in�nity and the splitting theorem from [Mon06] provides us with acanonical subspace X+×X− ⊆ X with component-wise and minimal H-action. All of ∂X+

is �xed by H−, which means that this boundary is empty. Since X is proper, it followsthat X+ is bounded and hence reduced to a point by minimality. Thus H+ �xes a point inX ⊆ X and co-amenability implies that G �xes a probability measure µ on X. If µ weresupported on ∂X, the proof of Proposition 7.1 would provide a G-�xed point at in�nity,which is absurd. Therefore µ(X) > 0. Now choose a bounded set B ⊆ X large enough sothat µ(B) > µ(X)/2. Then any G-translate of B must meet B. It follows that G has abounded orbit and hence a �xed point as claimed. �

Proof of Corollary 7.3. We refer to [CFP96] for a detailed introduction to the group F .In particular, F can be realised as the group of all orientation-preserving piecewise a�nehomeomorphisms of the interval [0, 1] that have dyadic breakpoints and slopes 2n withn ∈ Z. Given a subset A ⊆ [0, 1] we denote by FA < F the subgroup supported on A.

We claim that whenever A has non-empty interior, FA is co-amenable in F . The argumentis analogous to [MP03] and to [GM07, �4.F]; indeed, in view of the alternative de�nition ofF just recalled, one can choose a sequence {gn} in F such that gnA contains [1/n, 1− 1/n]and thus F gnA contains F[1/n,1−1/n]. Consider the compact space of means on F/FA, namely�nitely additive measures, endowed with the weak-* topology from the dual of `∞(F/FA).Any accumulation point µ of the sequence of Dirac masses at g−1

n FA will be invariant underthe union F ′ of the groups F[1/n,1−1/n]. Now F ′ is the kernel of the derivative homomorphismF → 2Z × 2Z at the pair of points {0, 1}. In particular, F ′ is co-amenable in F and thusthe F ′-invariance of µ implies that there is also a F -invariant mean on F/FA, which is oneof the characterisations of co-amenability [Eym72].

Let now X be any proper CAT(0) space with an F -action by isometries. We can assumethat F has no �xed point at in�nity and therefore we can also assume that X is minimal byProposition 3.1. The above claim provides us with many pairs of commuting co-amenablesubgroups upon taking disjoint sets of non-empty interior. Therefore, Corollary 7.2 showsthat X ∼= Rn for some n. In particular the isometry group is linear. Since F is �nitelygenerated (by g0 and g1 in the above presentation, compare also [CFP96]), Malcev's the-orem [Mal40] implies that the image of F is residually �nite. The derived subgroup of F(which incidentally coincides with the group F ′ introduced above) being simple [CFP96], itfollows that it acts trivially. It remains only to observe that two commuting isometries ofRn always have a common �xed point in Rn, which is a matter of linear algebra. �

The above reasoning can be adapted to yield similar results for branch groups and relatedgroups; we shall address these questions elsewhere.

7.B. Geometric density for subgroups of �nite covolume. The following geometricdensity theorem generalises Borel's density (see Proposition 7.8 below) and contains Theo-rem 1.19 from the Introduction.

Theorem 7.4. Let G be a locally compact group with a continuous isometric action on aproper CAT(0) space X without Euclidean factor.

If G acts minimally and without global �xed point in ∂X, then any closed subgroup with�nite invariant covolume in G still has these properties.

Remark 7.5. For a related statement without the assumption on the Euclidean factor ofX or on �xed points at in�nity, see Theorem 8.14 below.


Proof. Retain the notation of the theorem and let Γ < G be a closed subgroup of �niteinvariant covolume. In particular, Γ is co-amenable and thus has no �xed points at in�nityby Proposition 7.1. By Proposition 3.1, there is a minimal non-empty closed convex Γ-invariant subset Y ⊆ X and it remains to show Y = X. Choose a point x0 ∈ X and de�nef : X → R by

f(x) =∫G/Γ

(d(x, gY )− d(x0, gY )

)dg.

This integral converges because the integrand is bounded by d(x, x0). The function f iscontinuous, convex (by [BH99, II.2.5(1)]) and �quasi-invariant� in the sense that it satis�es

(7.i) f(hx) = f(x)− f(hx0) ∀h ∈ G.Since G acts minimally and without �xed point at in�nity, this implies that f is constant(see Section 2 in [AB98a]; alternatively, when ∂X is �nite-dimensional, it follows fromTheorem 1.6 since (7.i) implies that f is invariant under the derived subgroup G′).

In particular, d(x, gY ) is a�ne for all g. It follows that for all x ∈ X the closed set

Yx ={z ∈ X : d(z, Y ) = d(x, Y )

}is convex. We claim that it is parallel to Y in the sense that d(z, Y ) = d(y, Yx) for all z ∈ Yxand all y ∈ Y . Indeed, on the one hand d(z, Y ) is constant over z ∈ Yx by de�nition, and onthe other hand d(y, Yx) is constant by minimality of Y since d(·, Yx) is a convex Γ-invariantfunction. In particular, Yx is Γ-equivariantly isometric to Y via nearest point projection(compare [BH99, II.2.12]) and each Yx is Γ-minimal. At this point, Remarks 39 in [Mon06]show that there is an isometric Γ-invariant splitting

X ∼= Y × T.It remains to show that the �space of components� T is reduced to a point. Let thus s, t ∈ Tand let m be their midpoint. Applying the above reasoning to the choice of minimal set Y0

corresponding to Y ×{m}, we deduce again that the distance to Y0 is an a�ne function onX. However, this function is precisely the distance function d(·,m) in T composed with theprojection X → T . Being non-negative and a�ne on [s, t], it vanishes on that segment andhence s = t. �

Remark 7.6. When Γ is cocompact in G, the proof can be shortened by integrating justd(x, gY ) in the de�nition of f above.

Corollary 7.7. Let X be a proper CAT(0) space without Euclidean factor such that G =Is(X) acts minimally without �xed point at in�nity, and let Γ < G be a closed subgroup with�nite invariant covolume. Then:

(i) Γ has trivial amenable radical.(ii) The centraliser ZG(Γ) is trivial.(iii) If Γ is �nitely generated, then is has �nite index in its normaliser NG(Γ) and the

latter is a �nitely generated lattice in G.

Proof. (i) and (ii) follow by the same argument as in the proof of Theorem 1.6. For (iii)we follow [Mar91, Lemma II.6.3]. Since Γ is closed and countable, it is discrete by Baire'scategory theorem and thus is a lattice in G. Since it is �nitely generated, its automorphismgroup is countable. By (ii), the normaliser NG(Γ) maps injectively to Aut(Γ) and henceis countable as well. Thus NG(Γ), being closed in G, is discrete by applying Baire again.Since it contains the lattice Γ, it is itself a lattice and the index of Γ in NG(Γ) is �nite.Thus NG(Γ) is �nitely generated. �


As pointed out by P. de la Harpe, point(ii) implies in particular that any lattice in G isICC (which means by de�nition that all its non-trivial conjugacy classes are in�nite). As iswell known, this is the criterion ensuring that the type II1 von Neumann algebra associatedto the lattice is a factor [Tak02, �V.7].

Finally, we indicate why Theorem 7.4 implies the classical Borel density theorem of [Bor60].It su�ces to justify the following:

Proposition 7.8. Let k by a local �eld (Archimedean or not), G a semi-simple k-groupwithout k-anisotropic factors, X the symmetric space or Bruhat�Tits building associated toG = G(k) and L < G any subgroup. If the L-action on X is minimal without �xed point atin�nity, then L is Zariski-dense.

Proof. Let L be the (k-points of the) Zariski closure of L. Then L is semi-simple; thisfollows e.g. from a very special case of Corollary 4.8, which guarantees that the radical ofL is trivial.

In the Archimedean case, we may appeal to Karpelevich�Mostow theorem (see [Kar53]or [Mos55]): any semi-simple subgroup has a totally geodesic orbit in the symmetric space.So the only semi-simple subgroup acting minimally is G itself.

In the non-Archimedean case, we could appeal to E. Landvogt functoriality theorem [Lan00]which would �nish the proof. However, there is an alternative direct and elementary argu-ment which avoids appealing to loc. cit. and goes as follows. First notice that, by the sameargument as in the proof of Theorem 6.4(iv), the k-rank of a semi-simple subgroup actingminimally equals the k-rank of G (this holds in all cases, not only in the non-Archimedeanone). Therefore, the inclusion of spherical buildings BL → BG provided by the groupinclusion L → G has the property that BL is a top-dimensional sub-building of BG. Anelementary argument (see [KL06, Lemma 3.3]) shows that the union Y of all apartmentsof X bounded by a sphere in BL is a closed convex subset of X. Clearly Y is L-invariant,hence Y = X by minimality. Therefore BL = BG, which �nally implies that L = G. �

7.C. The limit set of subgroups of �nite covolume. Let X be a complete CAT(0)space and G a group acting by isometries on X. Recall that the limit set ΛG of G is theintersection of the boundary ∂X with the closure of the orbit G.x0 in X = X t ∂X of anyx0 ∈ X, this set being independent of x0.

Proposition 7.9. Let G be a locally compact group acting continuously by isometries on acomplete CAT(0) space X. If Γ < G is any closed subgroup with �nite invariant covolume,then ΛΓ = ΛG.

Consider the following immediate corollary, which in the special case of Hadamard man-ifolds follows from the duality condition, see 1.9.16 and 1.9.32 in [Ebe96].

Corollary 7.10. Let G be a locally compact group with a continuous action by isometrieson a proper CAT(0) space. If the G-action is cocompact, then any lattice in G has full limitset in ∂X. �

Proof of Proposition 7.9. We observe that for any non-empty open set U ⊆ G there is acompact set C ⊆ G such that U−1ΓC = G. Indeed, (using an idea of Selberg, compareLemma 1.4 in [Bor60]), it su�ces to take C so large that

µ(ΓC) > µ(Γ\G)− µ(ΓU),

where µ denotes an invariant measure on Γ\G; any right translate of ΓU in Γ\G will thenmeet ΓC.


Now let ξ ∈ ΛG and x0 ∈ X. For any neighbourhood V of ξ in ∂X, we shall construct anelement in ΛΓ ∩ V . Let U ⊆ G be a compact neighbourhood of the identity in G such thatUξ ⊆ V and let {gn} be a sequence of elements of G with gnx0 converging to ξ (one usesnets if X is not separable). In view of the above observation, there are sequences {un} inU and {cn} in C such that ungnc−1

n ∈ Γ. The points gnc−1n x0 remain at bounded distance

of gnx0 as n→∞, and thus converge to ξ. Therefore, choosing an accumulation point u of{un} in U , we see that uξ is an accumulation point of {ungnc−1

n x0}, which is a sequence inΓx0. �

For future use, we observe a variant of the above reasoning yielding a more precise factin a simpler situation:

Lemma 7.11. Let G be a locally compact group with a continuous cocompact action byisometries on a proper CAT(0) space X. Let Γ < G be a lattice and c : R+ → X a geodesicray such that G �xes c(∞). Then there is a sequence {γi} in Γ such that γic(i) remainsbounded over i ∈ N.

Proof. For the same reason as above, there is a compact set U ⊆ G such that G = UΓU−1.Choose now {gi} such that gic(i) remains bounded and write gi = uiγiv

−1i with ui, vi ∈ U .

We have

d(γic(i), c(0)) = d(givic(i), uic(0)) ≤ d(givic(i), gic(i)) + d(gic(i), uic(0))

≤ d(vic(i), c(i)) + d(gic(i), c(0)) + d(uic(0), c(0)).

This is bounded independently of i because d(vic(i), c(i)) ≤ d(vic(0), c(0)) since c(∞) isG-�xed. �

We shall also need the following:

Lemma 7.12. A locally compact group containing a �nitely generated subgroup whose clo-sure has �nite covolume is compactly generated.

Proof. Denoting the closure of the given �nitely generated subgroup by Γ, we can writeG = UΓC as in the proof of Proposition 7.9 with both U and C compact. Since Γ isa locally compact group containing a �nitely generated dense subgroup, it is compactlygenerated and the conclusion follows. �

8. CAT(0) lattices

8.A. Preliminaries on lattices. We begin this section with a few well known basic factsabout general lattices.

Proposition 8.1. Let G be a locally compact second countable group and N �G be a closednormal subgroup.

(i) Given a closed cocompact subgroup Γ < G, the projection of Γ on G/N is closed ifand only if Γ ∩N is cocompact in N .

(ii) Given a lattice Γ < G, the projection of Γ on G/N is discrete if and only if Γ ∩Nis a lattice in N .

Proof. See Theorem 1.13 in [Rag72]. �

The second well known result is straightforward to establish:


Lemma 8.2. Let G = H × D be a locally compact group. Given a lattice Γ < G and acompact open subgroup Q < D, the subgroup ΓQ := Γ∩ (H×Q) is a lattice in H×Q, whichis commensurated by Γ.

If moreover G/Γ is compact, then so is (H ×Q)/ΓQ. �

(As we shall see in Lemma 10.14 below, there is a form of converse.)

Let X be a proper CAT(0) space and G = Is(X) be its isometry group. Given a discretegroup Γ acting properly and cocompactly on X, then the quotient G\X is compact and theimage of Γ in G is a cocompact lattice (note that the kernel of the map Γ→ Is(X) is �nite).Conversely, if the quotient G\X is compact, then any cocompact lattice of G is a discretegroup acting properly and cocompactly on X.

Lemma 8.3. In the above setting, G is compactly generated and Γ is �nitely generated.

Proof. For lack of �nding a classical reference, we refer to Lemma 22 in [MMS04]). �

8.B. Variations on Auslander's theorem.

Lemma 8.4. Let A = RnoO(n) and S be a semi-simple Lie group without compact factor.Any lattice Γ in G = A × S has a �nite index subgroup Γ0 which splits as a direct productΓ0 ∼= ΓA × Γ′, where ΓA = Γ ∩ (A× 1) is a lattice in (A× 1).

Proof. Let V = Rn denote the translation subgroup of A and U denote the closure ofthe projection of Γ to S. The subgroup U < S is closed of �nite covolume; therefore it iseither discrete or it contains a semi-simple subgroup of positive dimension by Borel's densitytheorem (in fact one could be more precise using the Main Result of [Pra77], but this isnot necessary for the present purposes). On the other hand, Auslander's theorem [Rag72,Theorem 8.24] ensures that the identity component of the projection of Γ in S × A/V issoluble, from which it follows that U has a connected soluble normal subgroup. Thus U isdiscrete. Therefore, by Proposition 8.1, the group ΓA = Γ ∩ (A× 1) is a lattice in (A× 1).In particular ΓA is virtually Abelian [Thu97, Corollary 4.1.13].

Since the projection of Γ to S is a lattice in S, it is �nitely generated [Rag72, 6.18].Therefore Γ possesses a �nitely generated subgroup Λ containing ΓA and whose projectionto S coincides with the projection of Γ. Notice that Λ is a lattice in S × A by [Sim96,Theorem 23.9.3]; therefore Λ has �nite index in Γ, which shows that Γ is �nitely generated.

Since ΓA is normal in Γ, the projection ΓA of Γ to A normalises the lattice ΓA and is thusvirtually Abelian. Hence ΓA is a �nitely generated virtually Abelian group which normalisesΓA. Therefore ΓA has a �nite index subgroup which splits as a direct product of the formΓA×C, and the preimage Γ′ of C in Γ is a normal subgroup which intersects ΓA trivially. Inparticular the group Γ′ ·ΓA ∼= Γ′×ΓA is a �nite index normal subgroup of Γ, as desired. �

Lemma 8.5. Let Γ be a group containing a subgroup of the form Γ0 ∼= Γ0S×Γ0

A, where Γ0S is

isomorphic to a lattice in a semi-simple Lie group with trivial centre and no compact factor,and Γ0

A is amenable. If Γ commensurates Γ0, then Γ commensurates both Γ0S and Γ0

A.

Proof. Let Γ1 ∼= Γ1S×Γ1

A be a conjugate of Γ0 in Γ. The projection of Γ0∩Γ1 to Γ0S is a �nite

index subgroup of Γ0S . By Borel density theorem, it must therefore have trivial amenable

radical. In particular the projection of Γ0 ∩ Γ1A to Γ0

S is trivial. Therefore the image ofthe projection of Γ0 ∩ Γ1

S (resp. Γ0 ∩ Γ1A) to Γ0

S (resp. Γ0A) is of �nite index. The desired

assertion follows. �

Proposition 8.6. Let A = RnoO(n), S be a semi-simple Lie group with trivial centre andno compact factor, D be a totally disconnected locally compact group and G = S × A ×D.


Then any �nitely generated lattice Γ < G has a �nite index subgroup Γ0 which splits as adirect product Γ0

∼= ΓA × Γ′, where ΓA ⊆ Γ ∩ (1 × A × D) is a �nitely generated virtuallyAbelian subgroup whose projection to A is a lattice.

Proof. Let Q < D be a compact open subgroup. By Lemma 8.2, the intersection Γ0 =Γ ∩ (S × A × Q) is a lattice in S × A × Q, which is commensurated by Γ. Since Q iscompact, the projection of Γ0 to S × A is a lattice, to which we may apply Lemma 8.4.Upon replacing Γ0 by a �nite index subgroup (which amounts to replacing Q by an opensubgroup), this yields two normal subgroups Γ0

S ,Γ0A < Γ0 and a decomposition Γ0 = Γ0

S ·Γ0A,

where Γ0S ∩ Γ0

A ⊆ Q and Γ0A = Γ0 ∩ (1 × A × Q) is a �nitely generated virtually Abelian

group whose projection to A is a lattice.By virtue of Lemma 8.5, we deduce that the image of the projection of Γ to A com-

mensurates a lattice in A. But the commensurator of any lattice in A is virtually Abelian.Therefore, upon replacing Γ by a �nite index subgroup, it follows that the projection of Γto A normalises the projection of Γ0

A. We now de�ne

ΓA =⋂γ∈Γ

γΓ0Aγ−1.

Then the projection of ΓA coincides with the projection of Γ0A; in particular it is still a

lattice. Furthermore, the subgroup ΓA is normal in Γ. We now proceed as in the proofof Lemma 8.4. Since the projection of Γ to A is �nitely generated and virtually Abelian,we may thus �nd in this group a virtual complement to the image of the projection of ΓA.Let Γ′ be the preimage of this complement in Γ. Then, upon replacing Γ by a �nite indexsubgroup, the group Γ′ is normal in Γ and Γ = ΓA · Γ′. Since ΓA is normal as well, thecommutator [ΓA,Γ′] is contained in the intersection ΓA∩Γ′, which is trivial by construction.This �nally shows that Γ ∼= ΓA × Γ′, as desired. �

Remark 8.7. In the setting of Proposition 8.6, assume that any compact subgroup of Dnormalised by Γ is trivial. Then ΓA ⊆ 1×A×1 and the projection of Γ to S×D is discrete.Indeed, the de�nition of ΓA given in the proof shows that it is contained in 1×A× γQγ−1

for all γ ∈ Γ and under the current assumptions the intersection⋂γQγ−1 is trivial. The

claim about the projection to S ×D follows from Proposition 8.1.

8.C. Lattices, the Euclidean factor and �xed points at in�nity. Given a properCAT(0) space X and a discrete group Γ acting properly and cocompactly, it is a well knownopen question, going back to M. Gromov [Gro93, �6.B3], to determine whether the presenceof an n-dimensional �at in X implies the existence of a free Abelian group of rank n in Γ.(In the manifold case, see problem 65 on Yau's list [Yau82].) Here we propose the followingtheorem; the special case where X/Γ is a compact Riemannian manifold is the main resultof Eberlein's article [Ebe83] (compare also the earlier Theorem 5.2 in [Ebe80]).

Theorem 8.8. Let X be a proper CAT(0) space such that G = Is(X) acts cocompactly.Suppose that X ∼= Rn ×X ′.

(i) Any �nitely generated lattice Γ < G has a �nite index subgroup Γ0 which splits asa direct product Γ0

∼= Zn × Γ′.(ii) If moreover X is G-minimal ( e.g. if X is geodesically complete), then Zn acts

trivially on X ′ and as a lattice on Rn; the projection of Γ to Is(X ′) is discrete.

We recall that cocompact lattices are automatically �nitely generated in the above set-ting, Lemma 8.3. The following example shows that, without the assumption that G actsminimally, the projection of Γ to Is(X ′) should not be expected to have discrete image:


Example 8.9. Let X be the closed submanifold of R3 de�ned by X = {(x, y, z) ∈ R3 | 1 ≤z ≤ 2} and consider the following Riemannian metric on X:

ds2 = dx2 + z2dy2 + dz2.

One readily veri�es that it is non-positively curved; thus X is a CAT(0) manifold. Clearly Xsplits o� a one-dimensional Euclidean factor along the x-axis. Moreover the group H ∼= R2

of all translations along the xy-plane preserves X and acts cocompactly. Let Γ be thesubgroup of H generated by a and b, where

a : (x, y, z) 7→ (x, y, z) + (√

2, 1, 0) and b : (x, y, z) 7→ (x, y, z) + (1,√

2, 0).

Then Γ ∼= Z2 is a cocompact lattice in Is(X), but no non-trivial subgroup of Γ acts triviallyon the yz-factor of X. The projection of Γ to the isometry group of that factor is notdiscrete (see Proposition 8.1(ii)).

The above result is the converse to the Flat Torus Theorem when it is stated as in [BH99,II.7.1]. In particular we deduce that the dimension of the Euclidean de Rham factor is aninvariant of Γ. In the manifold case, again, this is the main point of [Ebe83].

Corollary 8.10. Let X be a proper CAT(0) space such that G = Is(X) acts cocompactlyand minimally. Let Γ < G be a �nitely generated lattice.

Then the dimension of the Euclidean factor of X equals the maximal rank of a free Abeliannormal subgroup of Γ.

In order to apply Theorem 1.1 and Addendum 1.4 towards Theorem 8.8, we will need thefollowing.

Theorem 8.11. Let X be a proper CAT(0) space such that G = Is(X) acts cocompactly andcontains a �nitely generated lattice. Then X contains a canonical closed convex G-invariantG-minimal subset X ′ 6= ∅ which has no Is(X ′)-�xed point at in�nity.

Consider the immediate corollary.

Corollary 8.12. Let X be a proper CAT(0) space such that G = Is(X) acts cocompactlyand minimally. If G contains a �nitely generated lattice, then G has no �xed point atin�nity. �

This shows that the mere existence of a �nitely generated lattice imposes restrictionson cocompact CAT(0) spaces; much more detailed results in that spirit will be given inSection 11.

We do not know whether the statement of Corollary 8.12 remains true without the �nitegeneration assumption on the lattice (see Problem 12.3 below).

Example 8.13. We emphasise that the full isometry group of a cocompact proper CAT(0)space may have global �xed points at in�nity; in fact, the space might even be homogeneous,as it is the case for E. Heintze's manifolds [Hei74] mentioned earlier. An even simpler wayto construct cocompact proper CAT(0) space with this property is to mimic Example 6.6:Start from a regular tree T , assuming for de�niteness that the valency is three. Replaceevery vertex by a congruent copy of an isosceles triangle that is not equilateral, in such a waythat its distinguished vertex always points to a �xed point at in�nity (of the initial tree).Then the stabiliser H in Is(T ) of that point at in�nity still acts faithfully and cocompactlyon the modi�ed space T ′; the construction is so that the isometry group of T ′ is in factreduced to H.


We shall also establish a strengthening of Corollary 8.12, which can be viewed as a formof Borel (or geometric) density theorem without assumption about �xed points at in�nity.

Theorem 8.14. Let X be a proper CAT(0) space such that G = Is(X) acts cocompactly andminimally. Assume there is a �nitely generated lattice Γ < G. Then Γ acts minimally onX and moreover all Γ-�xed points at in�nity are contained in the boundary of the (possiblytrivial) Euclidean factor of X.

We now turn to the proofs. In the case of a discrete cocompact group Γ = G, a version ofthe following was �rst established by Burger�Schroeder [BS87] (as pointed out in [AB98a,Corollary 2.7]).

Proposition 8.15. Let X be a proper CAT(0) space, G < Is(X) a closed subgroup whoseaction on X is cocompact and Γ < G a �nitely generated lattice. Then there exists a Γ-invariant closed convex subset Y ⊆ X which splits Γ-equivariantly as Y = E ×W , where Eis a (possibly 0-dimensional) Euclidean space on which Γ acts by translations and such that∂E contains the �xed point set of G in ∂X.

Proof. We can assume that there are G-�xed points at in�nity, since otherwise there isnothing to prove. We claim that for any G-�xed point ξ there is a geodesic line σ : R→ Xwith σ(+∞) = ξ such that any γ ∈ Γ moves σ to within a bounded distance of itself � andhence to a parallel line by convexity of the metric.

Indeed, let c : R+ → X be a geodesic ray with c(∞) = ξ and let {γi} be as in Lemma 7.11.Then, by Arzelà�Ascoli, there is a subsequence I ⊆ N and a geodesic line σ : R→ X suchthat σ(t) = limi∈I γi c(t + i) for all t. Since each g ∈ G has bounded displacement alongc, the sequence {γigγ−1

i }i∈I is bounded and thus we can assume that it converges for all g(recalling that G is second countable, but we shall only consider g ∈ Γ anyway). Since Γ isdiscrete and �nitely generated, we can further restrict I so that there is γ∞ ∈ Γ such that

γiγγ−1i = γ∞γγ

−1∞ ∀ γ ∈ Γ, i ∈ I.

Since

d(γγ−1∞ σ(t), γ−1

∞ σ(t)) = limi∈I

d(γ−1i γ∞γγ

−1∞ γic(t+ i), c(t+ i))

= limi∈I

d(γc(t+ i), c(t+ i)) ≤ d(γc(0), c(0)),

It now follows that every γ ∈ Γ has bounded displacement length along the geodesic γ−1∞ σ.

Thus the same holds for the geodesic σ which is therefore (by convexity) translated to aparallel line by each element of Γ as claimed.

Consider a �at E ⊆ X that is maximal for the property that each element of Γ hasconstant displacement length on E. Let Y be the union of all �ats that are at �nite distancefrom E. One shows that Y splits as Y ∼= E ×W for some closed convex W ⊆ X usingthe Sandwich Lemma [BH99, II.2.12] and Lemma II.2.15 of [BH99] just like in Section 2.Babove. The de�nition of Y shows that Γ preserves Y as well as its splitting and acts on theE coordinate by translations.

It remains to show that any G-�xed point ξ ∈ ∂X belongs to ∂E. First, ξ ∈ ∂Y since∂Y = ∂X by Corollary 7.10; we thus represent ξ by a ray c : R+ → Y . Let now σ bea geodesic line as provided by the claim. We can assume that σ lies in Y because it wasconstructed from Γ-translates of c and Y is Γ-invariant. One can write σ = (σE , σW ) whereσE , σW are linearly re-parametrised geodesics in E and W , see [BH99, I.5.3]. We need toprove that σW has zero speed. Since any given γ ∈ Γ has constant displacement along σ andon each of the parallel copies of E individually, its displacement is constant on the union of


all parallel copies of E visited by σ, which is E × σW (R). The latter being again a �at, themaximality of E shows that σW is constant. �

Proof of Theorem 8.11. Let Y = E × W ⊆ X be as in Proposition 8.15. Recall that∂Y = ∂X by Corollary 7.10. We claim that ∂X has circumradius > π/2. Indeed, it wouldotherwise have a G-�xed circumcentre by Proposition 2.1, but this circumcentre cannotbelong to ∂E since E is Euclidean; this contradicts Proposition 8.15. We now apply Corol-lary 2.10. This yields a canonical G-invariant closed convex subset X ′, which is minimalwith respect to the property that ∂X ′ = ∂X. It follows in particular by Corollary 7.10that Γ acts minimally on X ′. Let now X ′ = E′ ×X ′0 be the canonical splitting, where E′

is the maximal Euclidean factor [BH99, II.6.15]. On the one hand, since X ′ is Γ-minimal,Proposition 8.15 applied to X ′ shows that G has no �xed points in ∂X ′0 since E

′ is maximalas a Euclidean factor. On the other hand, Is(E′) �xes no point at in�nity on E′. We deducethat Is(X ′) ∼= Is(E′)× Is(X ′0) has indeed no �xed point at in�nity. �

End of proof of Theorem 8.14. Arguing as in the proof of Theorem 8.11, we establish thatX is Γ-minimal. Let X = X ′ × E be the canonical splitting, where E is the maximalEuclidean factor. Since any isometry of X decomposes uniquely as isometries of E and X ′

(II.6.15 in [BH99]), is su�ces to show that Γ has no �xed point in ∂X ′. This follows fromProposition 7.1 applied to the G-action on X ′. �

End of proof of Theorem 8.8. Assume �rst that X is G-minimal, recalling that this is thecase if X is geodesically complete by Lemma 2.13. In view of Corollary 8.12, we can applyTheorem 1.1 and we are therefore in the setting of Proposition 8.6. Since the group ΓAprovided by that proposition contains a �nite index subgroup isomorphic to Zn, we havealready established (i) under the additional minimality assumption.

In order to show (ii), it su�ces by Remark 8.7 to prove that any compact subgroup of Gnormalised by Γ is trivial. This follows from the fact that X is Γ-minimal, as established inTheorem 8.14.

It remains to prove (i) without the assumption that X is G-minimal. Let Y ⊆ X bethe G-minimal set provided by Theorem 8.11 and let Y ∼= Rm × Y ′ be its Euclideandecomposition. Then m ≥ n because of the characterisation of the Euclidean factor interms of Cli�ord isometries [BH99, II.6.15]; indeed, any (non-trivial) Cli�ord isometry ofX restricts non-trivially to Y because Y has �nite co-diameter. The kernel F � Γ of theΓ-action on Y is �nite and thus we can assume that it is central upon replacing Γ with a�nite index subgroup. Passing to a further �nite index subgroup, we know from the minimalcase that Γ/F splits as Γ/F = Zm × Λ′. Let ΓZm ,Γ′ � Γ be the pre-images in Γ of thosefactors. Thus we can write Γ = ΓZm · Γ′ with ΓZm ∩ Γ′ ⊆ F . It is straightforward that a�nite central extension of Zm is virtually Zm (see e.g. [BH99, II.7.9]). Therefore Γ containsa �nite index subgroup isomorphic to Zm × Γ′ and the result follows since m ≥ n. �

Proof of Corollary 8.10. Notice that a splitting Γ0∼= Zn×Γ′ with Γ0 normal and n maximal

provides a normal subgroup Zn � Γ since Zn is characteristic in Γ0. Therefore, givenTheorem 8.8, it only remains to see that a normal Zn � Γ of maximal rank forces X tohave a Euclidean factor of dimension at least n. Otherwise, the projection of Γ to the non-Euclidean factor X ′ would be a lattice by Theorem 8.8(ii) and contain an in�nite normalamenable subgroup, contradicting Corollary 7.7(i). �

Finally, we record that Theorem 1.21 is contained in Theorem 8.8 and Corollary 8.10for (i), and Corollary 8.12 and Theorem 8.14 for (ii).


8.D. Irreducible lattices in CAT(0) spaces. Recall from the Introduction that a (topo-logical) group is called irreducible if no (open) �nite index subgroup splits non-triviallyas a direct product of (closed) subgroups. For example, any locally compact group actingcontinuously, properly, minimally, without �xed point at in�nity on an irreducible properCAT(0) space is irreducible by Theorem 1.6.

In particular, an abstract group Γ is irreducible if it does not virtually split. This ter-minology is inspired by the concept of irreducibility for closed manifolds, which means thatno �nite cover of the manifold splits non-trivially. Of course, the universal cover of such amanifold can still split. Indeed, one gets many classical CAT(0) groups by considering �irre-ducible lattices� in products of simple Lie groups or more generally of semi-simple algebraicgroups over various local �elds.

The latter concept of irreducibility for lattices is de�ned as follows: A lattice Γ < G =G1× · · · ×Gn in a product of locally compact groups is called an irreducible lattice if itsprojections to any subproduct of the Gi's are dense and each Gi is non-discrete.

The point of this notion (and of the nearly confusing terminology) is that it preventsΓ and its �nite index subgroups from splitting as a product of lattices in Gi. Moreover,if all Gi's are centre-free simple Lie (or algebraic) groups without compact factors, theirreducibility of Γ as a lattice is equivalent to its irreducibility as a group in and for itself;this is a result of Margulis [Mar91, II.6.7]. As we shall see in Theorem 8.17 below, a versionof this equivalence holds for lattices in the isometry group of a CAT(0) space.

Remark 8.16.

(i) The non-discreteness of Gi is often omitted from this de�nition; the di�erence isinessential since the notion of a lattice is trivial for discrete groups. Notice howeverthat our de�nition ensures that all Gi are non-compact and that n ≥ 2.

(ii) One veri�es that any lattice Γ < G = G1 × G2 is an irreducible lattice in theproduct G∗ < G of the closures G∗i < Gi of its projections to Gi (provided theseprojections are non-discrete).

The following geometric version of Margulis' criterion contains Theorem 1.22 from theIntroduction.

Theorem 8.17. Let X be a proper CAT(0) space, G < Is(X) a closed subgroup actingcocompactly on X, and Γ < G a �nitely generated lattice.

(i) If Γ is irreducible as an abstract group, then for any �nite index subgroup Γ0 < Γand any Γ0-equivariant splitting X = X1 ×X2 with X1 and X2 non-compact, theprojection of Γ0 to both Is(Xi) is non-discrete.

(ii) If in addition the G-action is minimal, then the converse statement holds as well.

Remark 8.18. Recall that the G-minimality is automatic if X is geodesically complete(Lemma 2.13). Statement (ii) fails completely without minimality (as witnessed for instanceby the uncosmopolitan mien of an equivariant mane).

Proof of Theorem 8.17. Suppose Γ irreducible. Let X ′ ⊆ X be the canonical subspaceprovided by Theorem 8.11. By Theorem 8.8, the space X ′ has no Euclidean factor unlessX = R and Γ is virtually cyclic, in which case the desired statement is empty.

We �rst deal with the case when G acts minimally on X; by Theorem 7.4 this amountsto assume X = X ′. Suppose for a contradiction that for Γ0 and X ′ = X ′1 × X ′2 as inthe statement, the projection G1 of Γ0 to Is(X ′1) is discrete. Let G2 be the closure of theprojection of Γ0 to Is(X ′2) and notice that both Gi are compactly generated since Γ and


hence also Γ0 is �nitely generated. The projection Γ2 of Γ0 ∩ (1 × G2) to G2 is a lattice(by Lemma 8.2 or by Proposition 8.1); being normal, it is cocompact and hence �nitelygenerated. By Theorem 8.14, the group Γ0, and hence also G2, acts minimally and without�xed point at in�nity onX ′2. Therefore Corollary 7.7(ii) implies that the centraliser ZG2(Γ2)is trivial. But Γ2 is discrete, normal in G2, and �nitely generated. Hence ZG2(Γ2) is openand thus G2 is discrete. Therefore, the product G1 ×G2, which contains Γ0, is a lattice inIs(X ′1)× Is(X ′2) and thus in G. Now the index of Γ0 in G1 ×G2 is �nite and thus Γ0 splitsvirtually, a contradiction.

We now come back to the general case X ′ ⊆ X and suppose that X possesses a Γ0-equivariant splitting X = X1 ×X2. The group H = Is(X1) × Is(X2) < Is(X) contains Γ0;hence its action on X ′ is minimal without �xed point at in�nity by Corollary 8.12. There-fore, the splitting theorem [Mon06, Theorem 9] implies that X ′ possesses a Γ0-equivariantsplitting X ′ = X ′1×X ′2 induced by X = X1×X2 via H. Upon replacing Γ0 be a �nite indexsubgroup, the preceding paragraph thus yields a splitting Γ0/F ∼= G1 ×G2 of the image ofΓ0 in Is(X ′), where F denotes the kernel of the Γ0-action on X ′. Since F is �nite, so is theprojection to Is(X3−i) of the preimage Gi of Gi in Γ, for i = 1, 2. Therefore upon passing toa �nite index subgroup we may and shall assume that Gi acts trivially on Is(X3−i). Now thesubgroup of Is(X1)×Is(X2) generated by G1 and G2 splits as G1×G2 and is commensurableto Γ0, a contradiction.

Conversely, suppose now that the G-action is minimal and that Γ = Γ′ × Γ′′ splits non-trivially (after possibly having replaced it by a �nite index subgroup). If X = Rn, thenreducibility of Γ forces n ≥ 2 and we are done in view of the structure of Bieberbach groups.If X is not Euclidean but has a Euclidean factor, then Theorem 8.8(ii) provides a discreteprojection of Γ to the non-Euclidean factor Is(X ′); furthermore, X ′ is indeed non-compactas desired since otherwise by minimality it is reduced to a point, contrary to our assumption.

If on the other hand X has no Euclidean factor, then Γ acts minimally and without �xedpoint at in�nity by Theorem 8.11. Then the desired splitting is provided by the splittingtheorem [Mon06, Theorem 9]. Both projections of Γ are discrete, indeed isomorphic to Γ′

respectively Γ′′ because the cited splitting theorem ensures componentwise action of Γ. �

We now brie�y turn to uniquely geodesic spaces and to the analogues in this setting ofsome of P. Eberlein's results for Hadamard manifolds.

Theorem 8.19. Let X be a proper CAT(0) space with uniquely extensible geodesics suchthat Is(X) acts cocompactly on X.

If Is(X) admits a �nitely generated non-uniform irreducible lattice, then X is a symmetricspace (without Euclidean factor).

Proof. The action of Is(X) is minimal by Lemma 2.13 and without �xed point at in�nityby Corollary 8.12. Thus we can apply Theorem 1.1 and Addendum 1.4. Notice that Is(X)itself is non-discrete since it contains a non-uniform lattice; moreover, if it admits morethan one factor in the decomposition of Theorem 1.1, then the latter are all non-discrete byTheorem 8.17. Therefore, we can apply Theorem 6.10 to all factors of X. It remains onlyto justify that X has no Euclidean factor; otherwise, Auslander's theorem (compare alsoTheorem 8.8) implies X = R, which is incompatible with the fact that Γ is non-uniform. �

The following related result is due to P. Eberlein in the manifold case (Proposition 4.5in [Ebe82]). We shall establish another result of the same vein later without assuming thatgeodesics are uniquely extensible (see Theorem 11.6).


Theorem 8.20. Let X be a proper CAT(0) space with uniquely extensible geodesics andΓ < Is(X) be a discrete cocompact group of isometries. If Γ is irreducible (as an abstractgroup) and X is reducible, then X is a symmetric space (without Euclidean factor).

Proof. One follows line-by-line the proof of Theorem 8.19. The only di�erence is that, inthe present context, the non-discreteness of the isometry group of each irreducible factor ofX follows from Theorem 8.17 since X is assumed reducible. �

We can now conclude the proof of Theorem 1.27 from the Introduction. The �rst state-ment was established in Theorem 6.10. The second follows from Theorem 8.19 and the thirdfrom Theorem 8.20 in the uniform case, and from Theorem 8.19 in the non-uniform one. �

8.E. The hull of a lattice. Let X be a proper CAT(0) space X such that Is(X) actscocompactly on X. Let Γ < Is(X) be a �nitely generated lattice; note that the conditionof �nite generation is redundant if Γ is cocompact by Lemma 8.3. Theorem 8.11 providesa canonical Is(X)-invariant subspace X ′ ⊆ X such that G = Is(X ′) has no �xed point atin�nity.

In this section we shall de�ne the hull HΓ of the lattice Γ; this is a locally compact groupHΓ < Is(X ′) canonically attached to the situation and containing the image of Γ in Is(X ′).

For simplicity, we �rst treat the special case where Is(X) acts minimally; thus X ′ = Xand G = Is(X). Applying Theorem 1.1 and Addendum 1.4, we see in particular that Γpossesses a canonical �nite index normal subgroup Γ∗ = Γ ∩ G∗ which is the kernel of theΓ-action by permutation on the set of factors in the decomposition given by Addendum 1.4.

In the classical case when X is a symmetric space, the closure of the projection of Γ tothe isometry Is(Xi) of each factor in (1.ii) is an open subgroup of �nite index, as soon asX is reducible. This is no longer true in general, even in the case of Euclidean buildings.In fact, the same Γ may (and generally does) occur as lattice in increasingly large ambientgroups Γ < G < G′ < G′′ < · · · . In order to address this issue, we de�ne the hull as follows.Consider the closed subgroup HΓ∗ < G which is the direct product of the closure of theimages of Γ∗ in each of the factors in the decomposition (1.i) of Theorem 1.1. Then setHΓ = Γ ·HΓ∗ . In other words, we have inclusions

Γ < HΓ < G.

The closed subgroup HΓ∗ is nothing but the hull of the lattice Γ∗. It coincides withH∗Γ = HΓ ∩G∗. In particular H∗Γ = HΓ∗ is a direct product of irreducible groups satisfyingall the restrictions of Theorem 1.6 (except for the possible Euclidean motion factor), andthe image of Γ∗ in each of these factors is dense.

Remark 8.21. Notice that Γ is always a lattice in HΓ (by [Rag72, Lemma 1.6]). Weemphasise that HΓ is non-discrete and that Γ∗ is an irreducible lattice in HΓ∗ (in thesense of �8.D) as soon as Γ is irreducible as a group and X is reducible; this follows fromTheorem 8.17.

We now de�ne the hull HΓ < G in the general situation G = Is(X ′) with X ′ ⊆ X givenby Theorem 8.11. Since Is(X)\X is cocompact, it follows that X ′ is r-dense in X for somer > 0 and the canonical map Is(X) → G is proper. Let FΓ � Γ be the �nite kernel of theinduced map Γ → G and write Γ′ := Γ/FΓ. Then the hull of Γ is de�ned by HΓ := HΓ′

(reducing to the above case).In other words, Γ sits in HΓ only modulo the canonical �nite kernel FΓ. In fact, FΓ is

even canonically attached to Γ viewed as an abstract group.


Lemma 8.22. FΓ is a (necessarily unique) maximal �nite normal subgroup of Γ. Moreover,X ′ is Γ′-minimal.

Proof. The Γ′-action on X ′ is minimal by an application of Theorem 8.14 and thereforeevery �nite normal subgroup of Γ′ is trivial. Since moreover the Γ-action on X ′ is proper,it follows that a normal subgroup of Γ is �nite if and only if it lies in FΓ. �

For later references, we record the following expected fact.

Lemma 8.23. Assume that Γ is irreducible. If X ′ is reducible, then HΓ contains the identitycomponent of G := Is(X ′). In fact (HΓ)◦ = G◦ is a semi-simple Lie group with trivial centreand no compact factor.

Proof. By Theorem 8.8, the hypotheses on Γ imply that X ′ has no Euclidean factor. Thuseach almost connected factor of G∗ is a simple Lie group with trivial centre and no compactfactor. The projection of Γ∗ to each of these factors is non-discrete by Theorem 8.17 andthe assumption made on X ′. Its closure is semi-simple and Zariski dense by Theorem 7.4and Proposition 7.8. The result follows. �

8.F. On the canonical discrete kernel. Let G = G1 × G2 be a locally compact groupand Γ < G be an irreducible lattice. It follows from irreducibility that the projection to Giof the kernel of the projection Γ → Gj 6=i is a normal subgroup of Gi. In other words, wehave a canonical discrete normal subgroup Γi �Gi de�ned by

Γ1 = ProjG1

(Γ ∩ (G1 × 1)

)(and likewise for Γ2) which we call the canonical discrete kernel of Gi (depending on Γ).We observe that the image

Γ = Γ/(Γ1 · Γ2)of Γ in the canonical quotient G1/Γ1 ×G2/Γ2 is still an irreducible lattice (see Proposi-tion 8.1(ii)) and has the additional property that it projects injectively into both factors.

In this subsection, we collect some basic facts on lattices in (products of) totally discon-nected locally compact groups, adapting ideas of M. Burger and Sh. Mozes (see Proposi-tions 2.1 and 2.2 in [BM00b]).

Proposition 8.24. Let Γ < G = G1 × G2 be an irreducible lattice. Assume that G2 istotally disconnected, compactly generated and without non-trivial compact normal subgroup.If Γ is residually �nite, then canonical the discrete kernel Γ2 = Γ∩ (1×G2) commutes with

the discrete residual G(∞)2 .

Recall that the discrete residual G(∞) of a topological group G is by de�nition theintersection of all open normal subgroups. It is important to remark that, by Corollary 5.12the discrete residual of a non-discrete compactly generated locally compact group withoutnon-trivial compact normal subgroup is necessarily non-trivial.

Proof of Proposition 8.24. By a slight abuse of notation, we shall identify G2 with the sub-group 1×G2 of G. Given a �nite index normal subgroup Γ0 � Γ, the intersection Γ0 ∩Γ2 isa discrete normal subgroup of G2 (by irreducibility), contained as a �nite index subgroupin Γ2. Thus G2 acts by conjugation on the �nite quotient Γ2/Γ0 ∩ Γ2. In particular thekernel of this action is a �nite index closed normal subgroup, which is thus open. Therefore,the discrete residual G(∞)

2 acts trivially on Γ2/Γ0 ∩ Γ2. In other words, this means that

[Γ2, G(∞)2 ] ⊆ Γ0 ∩ Γ2.


Assume now that Γ is residually �nite. The preceding argument then shows that thecommutator [Γ2, G

(∞)2 ] is trivial, as desired. �

Proposition 8.25. Let Γ < G = G1 ×G2 be a cocompact lattice in a product of compactlygenerated locally compact groups. Assume that G2 is totally disconnected and that the cen-traliser in G1 of any cocompact lattice of G1 is trivial. If the discrete kernel Γ2 = Γ∩(1×G2)is trivial, then the quasi-centre QZ (G2) is topologically locally �nite.

Proof. Let S ⊆ QZ (G2) be a �nite subset of the quasi-centre. Then G2 possesses a compactopen subgroup U which centralises S. By Lemma 8.2 the group ΓU = Γ ∩ (G1 × U) isa cocompact lattice in G1 × U . In particular, there is a �nite generating set T ⊆ ΓU .By a lemma of Selberg [Sel60], the group ZΓ(T ) is a cocompact lattice in ZG(T ). ButZG(T ) = ZG(ΓU ) ⊆ 1×G2 since the projection of ΓU to G1 is a cocompact lattice. Sincethe discrete kernel Γ ∩ (1 × G2) is trivial by hypothesis, the centraliser ZΓ(T ) is trivialand, hence, ZG(T ) is compact. By construction S is contained in ZG(T ), which yields thedesired result. �

8.G. Residually �nite lattices.

Theorem 8.26. Let X be a proper CAT(0) space such that Is(X) acts cocompactly andminimally. Let Γ < Is(X) be a �nitely generated lattice. Assume that Γ is irreducible andresidually �nite. Then we have the following (see Section 8.E for the notation):

(i) Γ∗ acts faithfully on each irreducible factor of X.(ii) If Γ is cocompact and X is reducible, then for any closed subgroup G < Is(X)

containing HΓ∗ , we have QZ (G) = QZ (G∗) = 1. Furthermore soc(G∗) is a directproduct of r non-discrete closed subgroups, each of which is characteristically simple,where r is the number of irreducible factors of X.

(We emphasise that the irreducibility assumption concerns Γ as an abstract group; comparehowever Remark 8.21.)

Proof. If X is irreducible, there is nothing to prove. We assume henceforth that X isreducible. In view of Theorem 8.8, X has no Euclidean factor. Moreover, Corollary 8.12implies that Is(X) �xes no point at in�nity. In particular, Γ and HΓ∗ act minimally without�xed point at in�nity by Theorem 7.4.

Let H1, . . . ,Hr be the irreducible factors of HΓ∗ ; thus r coincides with the number ofirreducible factors of X. In view of Theorem 8.17, the group Γ∗ is an irreducible lattice inthis product. By Corollary 1.7 and Theorem 7.4, each Hi is either a centre-free simple Liegroup or totally disconnected with trivial amenable radical. If H1 is a simple Lie group,then it has no non-trivial discrete normal subgroup and hence

(Γ∗)1 := Γ∗ ∩ (H1 × 1× · · · × 1) = 1.

If H1 is totally disconnected, then by Proposition 8.24 the canonical discrete kernel (Γ∗)1

commutes with the discrete residual H(∞)1 , which is non-trivial by Corollary 5.12. Thus

ZH1(H(∞)1 ) = 1 by Theorem 1.6 and hence (Γ∗)1 = 1.

Assertion (i) now follows from a straightforward induction on r.

Assume next that Γ is cocompact. Let G1, . . . , Gr be the irreducible factors of G∗. ByLemma 8.23 and Proposition 8.25, and in view of Part (i), for each totally disconnected factorGi, the quasi-centre QZ (Gi) is topologically locally �nite. Its closure is thus amenable,hence trivial by Theorem 1.6. Moreover, the quasi-centre of each almost connected factoris trivial as well by Lemma 8.23.


Clearly the projection of the quasi-centre of G∗ to the irreducible factor Gi is containedin QZ (Gi). This shows that QZ (G∗) is trivial. Hence so is QZ (G), since it containsQZ (G∗) as a �nite index subgroup and since G has no non-trivial �nite normal subgroupby Corollary 4.8. Now the desired conclusion follows from Proposition 5.16. �

9. CAT(0) superrigidity

9.A. CAT(0) superrigidity for some classical non-uniform lattices. Let Γ be a non-uniform lattice in a simple (real) Lie group G of rank at least 2. By [LMR00, Theorem 2.15],unipotent elements of Γ are exponentially distorted. This means that, with respect to any�nitely generating set of Γ, the word length of |un| is an O(log n) when u is a unipotent.More generally an element u is called distorted if |un| is sublinear. If Γ is virtually boundedlygenerated by unipotent elements, one can therefore apply the following �xed point principle:

Lemma 9.1. Let Γ be a group which is virtually boundedly generated by distorted elements.Then any isometric Γ-action on a complete CAT(0) space such that elements of zero trans-lation length are elliptic has a global �xed point.

Proof. For any Γ-action on a CAT(0) space, the translation length of a distorted element iszero. Thus every such element has a �xed point; the assumption on Γ now implies that allorbits are bounded, thus providing a �xed point [BH99, II.2.8(1)]. �

Bounded generation is a strong property, which conjecturally holds for all (non-uniform)lattices of a higher rank semi-simple Lie group. It is known to hold for arithmetic groups insplit or quasi-split algebraic groups of a number �eld K of K-rank ≥ 2 by [Tav90], as wellas in a few cases of isotropic but non-quasi-split groups [ER06].

As noticed in a conversation with Sh. Mozes, Lemma 9.1 yields the following elementarysuperrigidity statement.

Proposition 9.2. Let Λ = SLn(Z[ 1p1···pk ]) with n ≥ 3 and pi distinct primes and set H =

SLn(Qp1)× · · · × SLn(Qpk).Given any isometric Λ-action on any complete CAT(0) space such that every element of

zero translation length is elliptic, there exists a Λ-invariant closed convex subspace on whichthe given action extends uniquely to a continuous H-action by isometries.

Proof. Let X be a complete CAT(0) space endowed with a Λ-action as in the statement.The subgroup Γ = SLn(Z) < Λ �xes a point by Lemma 9.1. The statement now followsbecause Γ is the intersection of Λ with the open subgroup SLn(Zp1)× · · · × SLn(Zpk) of H;for later use, we isolate this elementary fact as Lemma 9.3 below. �

Lemma 9.3. Let H be a topological group, U < H an open subgroup, Λ < H a densesubgroup and Γ = Λ ∩ U . Any Λ-action by isometries on a complete CAT(0) space witha Γ-�xed point admits a Λ-invariant closed convex subspace on which the action extendscontinuously to H.

Proof. Let X be the CAT(0) space and x0 ∈ X a Γ-�xed point. For any �nite subset F ⊆ Λ,let YF ⊆ X be the closed convex hull of Fx0. The closed convex hull Y of Λx0 is the closureof the union Y∞ of the directed family {YF }. Therefore, since the action is isometric andY is complete, it su�ces to show that the Λ-action on Y∞ is continuous for the topologyinduced on Λ by H. Equivalently, it su�ces to prove that all orbital maps Λ → Y∞ arecontinuous at 1 ∈ Λ. This is the case even for the discrete topology on Y∞ because thepointwise �xator of each YF is an intersection of �nitely many conjugates of Γ, the latterbeing open by de�nition. �


The same arguments as below show that Theorem 1.16 holds for any lattice of a higher-rank semi-simple Lie group which is boundedly generated by distorted elements (and ac-cordingly Theorem 1.17 generalises to suitable (S-)arithmetic groups).

Proof of Theorems 1.16 and 1.17. We start with the case Γ = SLn(Z). By Theorem 1.11,we obtain a closed convex subspace X ′ which splits as a direct product

X ′ ∼= X1 × · · · ×Xp × Y0 × Y1 × · · · × Yqin an Is(X ′)-equivariant way, where Y0

∼= Rn is the Euclidean factor. Each totally discon-nected factor Di of Is(X ′)∗ acts by semi-simple isometries on the corresponding factor Yi ofX ′ by Corollary 5.3. Therefore, by Lemma 9.1 for each i = 0, . . . , q, the induced Γ-actionon Yi has a global �xed point, say yi. In other words Γ stabilises the closed convex subset

Z := X1 × · · · ×Xp × {y0} × · · · × {yq} ⊆ X.Note that the isometry group of Z is an almost connected semi-simple real Lie group L.Combining Lemma VII.5.1 and Theorems VII.5.15 and VII.6.16 from [Mar91], it followsthat the Zariski closure of the image of Γ in L is a commuting product L1.L2, where L1

is compact, such that the corresponding homomorphism Γ → L2 extends to a continuoushomomorphism G→ L2. We de�ne Y ⊆ Z as the �xed point set of L1. Now L2, and henceΓ, stabilises Y . Therefore the continuous homomorphism G → L2 yields a G-action on Ywhich extends the given Γ-action, as desired.

Applying Theorem 6.4(iv) to the pair L2 < L acting on Z, we �nd in particular that L2

has no �xed point at in�nity in Y . Thus, upon replacing Y by a subspace, it is L2-minimal.Now Theorem 7.4 implies that the Γ- and G-actions on Y are minimal and without �xedpoint in ∂Y (although there might be �xed points in ∂X).

Turning to Theorem 1.17, the only change is that one replaces Lemma 9.1 by Proposi-tion 9.2. �

9.B. CAT(0) superrigidity for irreducible lattices in products. The aim of this sec-tion is to state a version of the superrigidity theorem [Mon06, Theorem 6] with CAT(0)targets. The original statement from loc. cit. concerns actions of lattices on arbitraryCAT(0) spaces, with reduced unbounded image. The following statement shows that, whenthe underlying CAT(0) space is nice enough, the assumption on the action can be consid-erably weakened.

Theorem 9.4. Let Γ be an irreducible uniform (or square-integrable weakly cocompact)lattice in a product G = G1×· · ·×Gn of n ≥ 2 locally compact σ-compact groups. Let X bea proper CAT(0) space with �nite-dimensional boundary. Given any Γ-action on X without�xed point at in�nity, if the canonical closed convex Γ-invariant Γ-minimal subset Y ⊆ Xprovided by Theorem 3.3(B.iii) has no Euclidean factor, then the Γ-action on Y extends toa continuous G-action by isometries.

Remark 9.5. Although the above condition on the Euclidean factor in the Γ-minimal sub-space Y might seem awkward, it cannot be avoided, as illustrated by Example 64 in [Mon06].Notice however that if Γ has the property that any isometric action on a �nite-dimensionalEuclidean space has a global �xed (for example if Γ has Kazhdan's property (T)), then anyminimal Γ-invariant subspace has no Euclidean factor.

Proof of Theorem 9.4. Let Y ⊆ X be the canonical subspace provided by Theorem 3.3(B.iii).Then Is(Y ) acts minimally on Y , without �xed point at in�nity. In particular we may applyTheorem 1.1 and Addendum 1.4. In order to show that the Γ-action on Y extends to a


continuous G-action, it is su�cient to show that the induced Γ-action on each irreduciblefactor of Y extends to a continuous G-action, factoring through some Gi. But the inducedΓ-action on each irreducible factor of Y is reduced by Corollary 2.8. Thus the result followsfrom [Mon06, Theorem 6]. �

9.C. Strong rigidity for product spaces. Superrigidity should contain, in particular,strong rigidity à la Mostow. This is indeed the content of Theorem 9.6 below, where anisomorphism of lattices is shown to extend to an ambient group. However, in contrast tothe classical case of symmetric spaces, which are homogeneous, the full isometry group doesnot in general determine the space since CAT(0) spaces are in general not homogeneous.Another di�erence is that the hull of a lattice, as described in Section 8.E, is generallysmaller than the full isometry group of the ambient CAT(0) space.

In view of the de�nition of the hull, the following statement is non-trivial only when X(or an invariant subspace) is reducible; this is expected since we want to use superrigidityfor irreducible lattices in products.

Theorem 9.6. Let X,Y be proper CAT(0) spaces and Γ,Λ discrete cocompact groups ofisometries of X, respectively Y , not splitting (virtually) a Zn factor.

Then any isomorphism Γ ∼= Λ determines an isomorphism HΓ∼= HΛ such that the fol-

lowing commutes:

Γ

��

// Λ

��

HΓ

∼= // HΛ

Theorem 9.6 provides a partial answer to Question 21 in [FHT08].

Remark 9.7. The assumption on Zn factors is equivalent to excluding Euclidean factorsfromX (or its canonical invariant subspace) by Theorem 8.8. On the one hand, this assump-tion is really necessary for the theorem to hold, even for symmetric spaces, since one cantwist the product using a Γ-action on the Euclidean factor when H1(Γ) 6= 0 (compare [LY72,�4]). On the other hand, since Bieberbach groups are obviously Mostow-rigid, Theorem 9.6together with Theorem 8.8 give us as complete as possible a description of the situationwith Zn factors.

Proof. Let X ′ ⊆ X be the subset provided by Theorem 8.11. We retain the notation FΓ �Γand Γ′ = Γ/FΓ < Is(X ′) from Section 8.E and recall from Lemma 8.22 that FΓ dependsonly on Γ as abstract group and that X ′ is Γ′-minimal. We de�ne Y ′, FΛ and Λ′ in the sameway and have the corresponding lemma. In particular, it follow that the isomorphism Γ ∼= Λdescends to Γ′ ∼= Λ′. Therefore, we can and shall assume from now on that X and Y areminimal and Γ < HΓ < Is(X), Λ < HΛ < Is(Y ). By Theorem 8.8, we know that X,Y haveno Euclidean factor. Thus Γ,Λ have no �xed point at in�nity by Theorem 8.14. We claimthat Γ has a �nite index subgroup Γ† which decomposes as a product Γ† = Γ1 × · · · × Γsof irreducible factors, with s maximal for this property. Indeed, otherwise we could applythe splitting theorem of [Mon06] to a chain a �nite index subgroups and contradict theproperness of X. We write Λ† = Λ1×· · ·×Λs for the corresponding groups in Λ. Combiningthe splitting theorem with Addendum 1.4, it follows from the de�nition of the hull that itis su�cient to prove the statement for s = 1. We assume henceforth that Γ, and hence alsoΛ, is irreducible. Furthermore, if X and Y are both irreducible, then HΓ = Γ and HΛ = Λand the desired statement is empty. We now assume that X is reducible.


By Theorem 7.4, the lattice Γ (resp. Λ) acts minimally without �xed point at in�nityon X (resp. Y ). Theorem 9.4 yields a continuous morphism f : HΓ∗ → HΛ∗ , which showsin particular (by the splitting theorem [Mon06]) that Y is reducible as well. A secondapplication of Theorem 9.4 yields a second continuous morphism f ′ : HΛ∗ → HΓ∗ . Noticethat the respective restrictions to Γ∗ and Λ∗ coincides with the given isomorphism and itsinverse. In particular f ′ ◦ f (resp. f ′ ◦ f) is the identity on Γ (resp. Λ). By de�nition ofthe hull, it follows that f ′ ◦ f (resp. f ′ ◦ f) is the identity on HΓ∗ (resp. HΛ∗). The desiredresult �nally follows, since there is a canonical isomorphism Γ/Γ∗ ∼= HΓ/HΓ∗ and since theaction of HΓ/HΓ∗ on HΓ∗ is canonically determined by the action of Γ/Γ∗ on Γ∗. �

The above proof shows in particular that amongst spaces that are Γ-minimal withoutEuclidean factor, the number of irreducible factors depends only upon the group Γ. If wecombine this with Theorem 8.14, Corollary 8.10 and Theorem 8.8(ii), we obtain that thenumber of factors in the �de Rham� decomposition (1.ii) is an invariant of the group:

Corollary 9.8. Let X be a proper CAT(0) space and Γ < Is(X) be a group acting properlydiscontinuously and cocompactly.

Then any other such space admitting a proper cocompact Γ-action has the same numberof factors in (1.ii) and the Euclidean factor has same dimension. �

(We recall that minimality is automatic when X is geodesically complete: Lemma 2.13.)

10. Arithmeticity of abstract lattices

The main goal of this section is to prove Theorem 1.29, which we now state in a slightlymore general form. Following G. Margulis [Mar91, IX.1.8], we shall say that a simplealgebraic group G de�ned over a �eld k is admissible if none of the following holds:

� char(k) = 2 and G is of type A1, Bn, Cn or F4.� char(k) = 3 and G is of type G2.

A semi-simple group will be said admissible if all its factors are.


If Γ admits a faithful Zariski-dense representation in an admissible semi-simple group(over any �eld), then the amenable radical R of G is compact and the quasi-centre QZ (G)is virtually contained in Γ · R. Furthermore, upon replacing G by a �nite index subgroup,the quotient G/R splits as G+ ×QZ (G/R) where G+ is a semi-simple algebraic group andthe image of Γ in G+ is an arithmetic lattice.

Since the projection map G→ G/R is proper, the statement of Theorem 10.1 implies inparticular that QZ (G/R) is discrete.

Corollary 10.2. Let G = G1×· · ·×Gn be a product of locally compact groups. Assume thatG admits a �nitely generated irreducible lattice with a faithful Zariski-dense representationin a semi-simple group over some �eld of characteristic 6= 2, 3.

Then G is a compact extension of a direct product of a semi-simple algebraic group by adiscrete group. �

To be more precise, the arithmeticity conclusion of Theorem 10.1 means the following.

There exists a global �eld K, a connected semi-simple K-anisotropic K-group H and a�nite set Σ of valuations of K such that:

(i) The quotient Γ := Γ/Γ ∩ (R ·QZ (G)) is commensurable with the arithmetic groupH(K(Σ)), where K(Σ) is the ring of Σ-integers of K. Moreover, Σ contains all Archimedean


valuations v for which H isKv-isotropic, whereKv denotes the v-completion ofK. In partic-ular, by Borel�Harish-Chandra and Behr�Harder reduction theory, the diagonal embeddingrealises H(K(Σ)) as a lattice in the product

∏v∈Σ H(Kv).

(ii) The group G+ is isomorphic to∏v∈Σ H(Kv)+ and this isomorphism implements the

commensurability of Γ with H(K(Σ)). For background references, including on H(Kv)+,see [Mar91, I.3].

In contrast to statements in [Mon05], there is no assumption on the subgroup structureof the factors Gi in Theorem 10.1, which may not even be irreducible. The nature of thelinear representation is however more restricted.

Another improvement is that no (weak) cocompactness assumption is made on Γ. Inparticular, under the same algebraic restrictions on the factors Gi as in loc. cit., we obtainthe following arithmeticity vs. non-linearity alternative for all �nitely generated lattices.

Corollary 10.3. Let Γ < G = G1 × · · · × Gn be an irreducible �nitely generated lattice,where each Gi is a locally compact group such that every non-trivial closed normal subgroupis cocompact. Then one of the following holds:

(i) Every �nite-dimensional linear representation of Γ in characteristic 6= 2, 3 has vir-tually soluble image.

(ii) G is a semi-simple algebraic group and Γ is an arithmetic lattice.

The hypothesis made on each factor Gi may be used to describe to some extent itsstructure independently of the existence of a lattice in G, see Proposition 5.13; in particulareach Gi is monolithic. However, we will not appeal to this preliminary description of theGi when proving Corollary 10.3: the structural information will instead be obtained aposteriori.

Remark 10.4. In [Mon05], the conclusion (i) was replaced by �niteness of the image.This follows from the current conclusion in the more restricted setting of loc. cit. thanksto Y. Shalom's superrigidity for characters [Sha00], unless of course Gi admits (virtually)a non-zero continuous homomorphism to R (after all in the current setting we can haveGi = R). It is part of the assumptions in [Mon05] that no such homomorphism exists, sothat Corollary 10.3 indeed generalises loc. cit.

10.A. Superrigid pairs. For convenience, we shall use the following terminology. Let J bea topological group and Λ < J any subgroup. We call the pair (Λ, J) superrigid if for anylocal �eld k and any connected absolutely almost simple adjoint k-group H, every abstracthomomorphism Λ → H(k) with unbounded Zariski-dense image extends to a continuoushomomorphism of J .

Proposition 10.5. Let (Λ, J) be a superrigid pair with J locally compact and Λ �nitelygenerated with closure of �nite covolume in J .

If Λ admits a faithful representation in an admissible semi-simple group (over any �eld)with Zariski-dense image, then the amenable radical R of J is compact and the quasi-centreQZ (J) is virtually contained in Λ · R. Furthermore, upon replacing J by a �nite indexsubgroup, the quotient J/R splits as J+ ×QZ (J/R) where J+ is a semi-simple algebraicgroup and the image of Λ in J+ is an arithmetic lattice.

(We point out again that in particular the direct factor QZ (J/R) is discrete.)

One might expect that Theorem 10.1 could now be proved by establishing in completegenerality that �nitely generated irreducible lattices in products of locally compact groups


form a superrigid pair. For uniform lattices, or more generally weakly cocompact square-summable lattices, this is indeed true and was proved in [Mon06]. We do not have aproof in general and shall eschew this di�culty by giving �rst an independent proof ofthe compactness of the amenable radical (Corollary 10.13 below) and using the residual�niteness of �nitely generated linear groups before proceeding with Proposition 10.5.

Nevertheless, we do have a general proof as soon as the groups are totally disconnected.


If G is totally disconnected, then (Γ, G) is a superrigid pair.

(As we shall see in Proposition 10.10, one can drop the �nite generation assumption in thesimpler case where Γ projects faithfully to some factor Gi.)

Proof of Proposition 10.5. We will largely follow the ideas of Margulis, deducing arith-meticity from superrigidity [Mar91, Chapter IX]. It is assumed that the reader has a copyof [Mon05] at hand, since it contains a similar reasoning under di�erent hypotheses. Thecharacteristic assumption in loc. cit. will be replaced by the current admissibility assump-tion.

The group J (and hence also all �nite index subgroups and factors) is compactly generatedby Lemma 7.12. Let τ : Λ→ H be the given faithful representation. Upon replacing Λ andJ by �nite index subgroups and post-composing τ with the projection map H→ H/Z (H),we shall assume henceforth that H is adjoint and Zariski-connected. The representationτ : Λ→ H need no longer be faithful, but it still has �nite kernel. As in [Mon05, (3.3)], inview of the assumption that Λ is �nitely generated, we may assume that H is de�ned over a�nitely generated �eld K. This is the �rst of two places where the admissibility assumptionis used in loc. cit. (following VIII.3.22 and IX.1.8 in [Mar91]).

By Tits' alternative [Tit72], the amenable radical of Λ is soluble-by-locally-�nite and thuslocally �nite since τ(Λ) is Zariski-dense and H is semi-simple. The �nite generation of Kimplies that this radical is in fact �nite (see e.g. Corollary 4.8 in [Weh73]), thus trivialby Zariski-density since H is adjoint. (The �nite generation of K is essential in positivecharacteristic since for algebraically closed �elds there is always a locally �nite Zariski-densesubgroup [BGM04].)

It now follows that if J is a compact extension of a discrete group, then the latterhas trivial amenable radical and thus all the conclusions of Proposition 10.5 hold trivially.Therefore, we assume henceforward that J is not compact-by-discrete.

Let H = H1× · · · ×Hk be the decomposition of H into its simple factors. We shall workwith the factors Hi one at a time. Let τi : Λ → Hi be the induced representation of Λ.Notice that τi need not be faithful; however, it has Zariski-dense (and in particular in�nite)image.

We let Σi denote the set of all (inequivalent representatives of) valuations v of K suchthat the image of τi(Λ) is not relatively compact in Hi(Kv) (for the Hausdor� topology);observe that this image is still Zariski-dense. Then Σi is non-empty since τi(Λ) is in�nite,see [BG07, Lemma 2.1].

By hypothesis, there exists a continuous representation J → Hi(Kv) for each v ∈ Σi,extending the given Λ-representation. We denote by Nv�J the kernel of this representation.Let I ⊆ {1, . . . , k} be the set of all those indices i such that J/Nv is non-discrete for eachv ∈ Σi.

We claim that the set I is non-empty.


Indeed, for each index j 6∈ I, there exists vj ∈ Σj such that Nvj is open in J . Thus thekernel

J+ =⋂j 6∈I

Nvj

of the continuous representation J →∏j 6∈I Hj(Kvj ) is open.

By assumption the closure of Λ in J has �nite covolume. Therefore, for each opensubgroup F < J , the closure of Λ∩F has �nite covolume in F . It follows in particular thatΛ ∩ F is in�nite unless F is compact.

These considerations apply to the open subgroup J+ < J . Since J is not compact-by-discrete, we deduce that Λ∩J+ is in�nite. Therefore the restriction to Λ of the representationJ →

∏j 6∈I Hj(Kvj ) has in�nite kernel and, hence, it does not factor through τ : Λ→ H(K).

In particular it cannot coincide with the given representation τ : Λ → H. Thus I is non-empty.

We claim that for each i ∈ I, the set Σi is �nite.

Let i ∈ I and v ∈ Σi. The arguments of [Mon05, (3.7)] show that the isomorphicimage of J/Nv in Hi(Kv) contains Hi(Kv)+. These arguments use again the admissibilityassumption because the appeal to a result of R. Pink [Pin98]; the fact that the latterhold in the admissible case is explicit in the table provided in Proposition 1.6 of [Pin98].Furthermore, it follows from Tits' simplicity theorem [Tit64] combined with [BT73, 6.14]that each J/Nv is quasi-simple in the sense of the de�nition made on p. 28. Moreover,an application of [BT73, 8.13] shows that the various quotients (J/Nv)v∈Σi are pairwisenon-isomorphic. In particular the normal subgroups (Nv)v∈Σi are pairwise distinct.

Let Di =⋂v∈Σi

Nv and recall that J/Di is compactly generated. Projecting each Nv toJ/Di, we obtain a family of pairwise distinct normal subgroups of J/Di indexed by Σi suchthat each corresponding quotient is quasi-simple, non-discrete and non-compact. Therefore,the desired claim follows from Corollary 5.15.

In particular, appealing again to [BT73, Corollaire 8.13], we obtain a continuous mapJ →

∏v∈Σi

Hi(Kv) which we denote again by τJi . The kernel of τJi is Di. Upon replacing

J and Λ by �nite index subgroups we may assume that the image of τJi coincides in factwith

∏v∈Σi

Hi(Kv)+, compare [Mon05, (3.9)].

We claim that R := J+ ∩ D is compact and that J = J+ · D, where D is de�ned byD =

⋂i∈I Di.

We �rst show that R = J+ ∩D is compact. Assume for a contradiction that this is notthe case. Then, given a compact open subgroup U of J , the intersection Λ0 = Λ ∩ (U · R)is in�nite: this follows from the same argument as above, using the assumption that theclosure of Λ has �nite covolume.

For each index j 6∈ I, we have J+ ⊆ Nvj and we deduce that the image of Λ0 in Hj(Kvj )is �nite, since it is contained in the image of U . Equivalently, the subgroup τj(Λ0) < Hj(K)is �nite. It follows in particular that τi(Λ0) is in�nite for some i ∈ I. By [BG07, Lemma 2.1],there exists v ∈ Σi such that the image of Λ0 in Hi(Kv) is unbounded. This is absurd sinceD ⊆ Nv and hence the image of Λ0 in Hi(Kv) is contained in the image of the compactsubgroup U . This shows that the intersection R is indeed compact.

At this point we know that the quotient J/D is isomorphic to a subgroup of the product∏i∈I

∏v∈Σi

Hi(Kv)+


which projects surjectively onto each factor of the form∏v∈Σi

Hi(Kv)+. Using again theGoursat-type argument as in Corollary 5.15, we �nd that J/D is indeed isomorphic toa �nite product of non-compact non-discrete simple groups Hi(Kv)+. In particular thequotient J/D has no non-trivial open normal subgroup. Since J+ is open and normal in J ,we deduce that J = J+ ·D, thereby establishing the claim.

By the very nature of the statement, we may replace J by the quotient J/R without anyloss of generality, since R is compact. In view of this further simpli�cation, the precedingclaim implies that J ∼= J+ ×D. In particular D is discrete.

It now follows as in [Mon05, (3.11)] that K is a global �eld, and that the image of Λ inthe semi-simple group J/D is an arithmetic lattice (compare [Mon05, (3.13)]). Therefore,by Proposition 8.1, the intersection Λ ∩D is a lattice in D and, hence, the discrete normalsubgroup D is virtually contained in Λ. As J ∼= J+ ×D and J+ has trivial quasi-centre, itfollows that the quasi-centre of J coincides with D. This �nishes the proof. �

For later use, we single out a (simpler) version of an argument referred to above.

Lemma 10.7. Let H be an admissible connected absolutely almost simple adjoint k-groupH, where k is a local �eld. Let J be a locally compact group with a continuous unboundedZariski-dense homomorphism τ : J → H(k). Then any compact normal subgroup of J iscontained in the kernel of τ .

Proof. Let K�J be a compact normal subgroup. The Zariski closure of τ(K) is normalisedby the Zariski-dense group τ(J) and therefore it is either H(k) or trivial. We assume theformer since otherwise we are done.

We claim that we can assume k non-Archimedean. Otherwise, either k = R or k = C. Inthe �rst case, τ(K) coincides with its Zariski closure by Weyl's algebraicity theorem [Vin94,4.2.1] so that H(k) is compact in which case the lemma is void by the unboundednessassumption. In the second case, one can reduce to the case τ(J) ⊆ H(R) as in [Mon05,(3.5)] and thus τ(K) = 1 as before since H(R) is also simple; the claim is proved.

Following now an idea from [Sha00, p. 41] (see also the explanations in Section (3.7)of [Mon05]), one uses [Pin98] to deduce that τ(K) is open upon possibly replacing k bya closed sub�eld (the admissibility assumption enters as in the proof of Proposition 10.5).We can still denote this sub�eld by k because it accommodates the whole image τ(J), seeagain [Mon05, (3.7)]. Now τ(J) is an unbounded open subgroup and hence contains H(k)+

by a result of J. Tits (see [Pra82]; this also follows from the Howe�Moore theorem [HM79]which however is posterior to Tits' result). This implies that the compact group τ(K) istrivial since H(k)+ is simple by [Tit64]. �

10.B. Boundary maps. We record two statements extracted from Margulis' work in theform most convenient for us.

Proposition 10.8. Let J be a second countable locally compact group with a measure classpreserving action on a standard probability space B. Let Λ < J be a dense subgroup witha Zariski-dense unbounded representation τ : Γ → H(k) to a connected absolutely almostk-simple adjoint group H over an arbitrary local �eld k.

If there is a proper k-subgroup L < H and a Λ-equivariant non-essentially-constant mea-surable map B → H(k)/L(k), then τ extends to a continuous homomorphism J → H(k).

Proof. The argument is given by A'Campo�Burger in the characteristic zero case at the endof Section 7 in [AB94] (pp. 18�19). This reference considers homogeneous spaces for B but


this restriction is never used. The general statement is referred to in [Bur95] and details aregiven in [Bon04]. �

Proposition 10.9. Let Γ be a countable group with a Zariski-dense unbounded representa-tion Γ→ H(k) to a connected absolutely almost k-simple adjoint group H over an arbitrarylocal �eld k. Let B be a standard probability space with a measure class preserving Γ-actionthat is amenable in Zimmer's sense [Zim84] and such that the diagonal action on B2 isergodic.

Then there is a proper k-subgroup L < H and a Γ-equivariant non-essentially-constantmeasurable map B → H(k)/L(k).

Proof. Again, this is proved in [AB94] for the characteristic zero case (and B homogeneous)and the necessary adaptations to the general case are explained in [Bon04]. �

We shall need these speci�c statements below. They �rst appeared within the proofof Margulis' commensurator superrigidity, which can adapted as follows using [Bur95] andLemma 9.3, providing a �rst step towards Theorem 10.6.

Proposition 10.10. Let G = G1 × G2 be a product of locally compact σ-compact groupsand Λ < G be an irreducible lattice. Assume that the projection of Λ to G1 is injective andthat G2 admits a compact open subgroup. Then the pair (Λ, G) is superrigid.

Proof. We claim that one can assume G second countable. As explained in [Mon06, Proposi-tion 61], σ-compactness implies the existence of a compact normal subgroup K�G meetingΛ trivially and such that G/K is second countable. Applying the statement to G/K to-gether with the image of Λ therein yields the general statement since the projection of Λ toG/K is an isomorphism; this proves the claim.

Let τ : Λ→ H(k) be as in the de�nition of superrigid pairs and let U < G2 be a compactopen subgroup. Set ΛU = Λ∩ (G1 ×U). By the injectivity assumption and Lemma 8.2, wecan consider ΛU as a lattice in G1 which is commensurated by (the image of the projectionof) Λ. We distinguish two cases.

Assume �rst that τ(ΛU ) is unbounded in the locally compact group H(k). We may thenapply Margulis' commensurator superrigidity in its general form proposed by M. Burger [Bur95,Theorem 2.A], see [Bon04] for details. This yields a continuous map J → H(k) factoringthrough G1 and extending the given Λ-representation, as desired.

Assume now that τ(ΛU ) is bounded, which is equivalent to ΛU �xing a point in thesymmetric space or Bruhat�Tits building associated to H(k). Then Lemma 9.3 yields acontinuous map J → H(k) factoring through G2. �

10.C. Radical superrigidity.

Theorem 10.11. Let G be a locally compact group, R � G its amenable radical, Γ < G a�nitely generated lattice and F the closure of the image of Γ in G/R.

Then any Zariski-dense unbounded representation of Γ in any connected absolutely almostsimple adjoint k-group H over any local �eld k arises from a continuous representation ofF via the map Γ→ F .

(In particular, the pair (Γ/(Γ ∩R), F ) is superrigid.)

Proof. Notice that G is σ-compact since it contains a �nitely generated, hence countable,lattice. (In fact G is even compactly generated by Lemma 7.12.) Set J = G/R. Thereexists a standard probability J-space B on which the Γ-action is amenable and such thatthe diagonal Γ-action on B2 is ergodic; it su�ces to choose B to be the Poisson boundary


of a symmetric random walk with full support on J . Indeed: (i) The J-action is amenableas was shown by Zimmer [Zim78]; this implies that the G-action is amenable since R is anamenable group and thus that the Γ-action is amenable since Γ is closed in G (see [Zim84,5.3.5]). (ii) The diagonal action of any closed �nite covolume subgroup F < J on B2 isergodic in view of the ergodicity with coe�cients of J , and hence the same holds for densesubgroups of F . For detailed background on this strengthening of ergodicity introducedin [BM02] and on the Poisson boundary in general, we refer the reader to [Ka��03].

Let now k be a local �eld, H a connected absolutely almost simple k-group and Γ→ H(k)a Zariski-dense unbounded representation. We can apply Proposition 10.9 and obtain aproper subgroup L < H and a Γ-equivariant map B → H(k)/L(k). Writing Λ for theimage of Γ in J , we can therefore apply Proposition 10.8 with F instead of J and theconclusion follows. �

Remark 10.12. An examination of this proof shows that one has also the following relatedresult. Let J be a second countable locally compact group and Λ ⊆ J a dense countablesubgroup whose action on J by left multiplication is amenable. Then the pair (Λ, J) issuperrigid. Indeed, one can again argue with Propositions 10.8 and 10.9 because it is easyto check that in the present situation any amenable J-space is also amenable for Λ viewedas a discrete group. Related ideas were used by R. Zimmer in [Zim87].

Corollary 10.13. Let G be a locally compact group and Γ < G a �nitely generated lattice.If Γ admits a faithful Zariski-dense representation in an admissible semi-simple group

(over any �eld), then the amenable radical of G is compact.

Proof. Let R be the amenable radical of G, F be the closure of the image of Γ in G/R andJ < G the preimage of F in G. The content of Theorem 10.11 is that the pair (Γ, J) issuperrigid. Since in addition Γ is closed and of �nite covolume in J (see [Rag72, Lemma 1.6]),we may apply Proposition 10.5 and deduce that the amenable radical of J is compact. Theconclusion follows since R < J . �

10.D. Lattices with non-discrete commensurators. The following useful trick allowsto realise the commensurator of any lattice in a locally compact group G as a lattice in aproduct G × D. A similar reasoning in the special case of automorphism groups of treesmay be found in [BG02, Theorem 6.6].

Lemma 10.14. Let Λ be a group and Γ < Λ a subgroup commensurated by Λ. Let Dbe the completion of Λ with respect to the left or right uniform structure generated by theΛ-conjugates of Γ. Then D is a totally disconnected locally compact group.

If furthermore G is a locally compact group containing Λ as a dense subgroup such thatΓ is discrete (resp. is a lattice) in G, then the diagonal embedding of Λ in G×D is discrete(resp. is an irreducible lattice).

The above lemma is in some sense a converse to Lemma 8.2. In the special case where onestarts with a lattice satisfying a faithfulness condition, this relation becomes even stronger.

Lemma 10.15. Let G,H be locally compact groups and Λ < G×H a lattice. Assume thatthe projection of Λ to G is faithful and that both projections are dense. Let U < H be acompact open subgroup, set Γ = Λ∩ (G×U) as in Lemma 8.2 and consider the group D asin Lemma 10.14 (upon viewing Λ as a subgroup of G). De�ne the compact normal subgroupK �H as the core K =

⋂h∈H hUh

−1 of U in G.Then the map Λ→ D induces an isomorphism of topological groups H/K ∼= D.


Proof of Lemma 10.14. One veri�es readily the condition given in [Bou60] (TG III, �3, No 4,Théorème 1) ensuring that the completion satis�es the axioms of a group topology. Weemphasise that it is part of the de�nition of the completion that D is Hausdor�; in otherwords D is obtained by �rst completing Λ with respect to the group topology as de�nedabove, and then dividing out the normal subgroup consisting of those elements which arenot separated from the identity.

Let U denote the closure of the projection of Γ to D. By de�nition U is open. Noticethat it is compact since it is a quotient of the pro�nite completion of Γ by construction. Inparticular D is locally compact.

By a slight abuse of notation, let us identify Γ and Λ with their images inD. We claim thatU ∩ Λ = Γ. Indeed, let {γn}n≥0 be a sequence of elements of Γ such that limn γn = λ ∈ Λ.Since λΓλ−1 is a neighbourhood of the identity in Λ (with respect to the topology inducedfrom D), it follows that γnλ−1 ∈ λΓλ−1 for n large enough. Thus λ ∈ γnΓ = Γ.

Assume now that Γ is discrete and choose a neighbourhood V of the identity in G suchthat Γ ∩ V = 1. In view of the preceding claim the product V × U is a neighbourhood ofthe identity in G×D which meets Λ trivially, thereby showing that Λ is discrete.

Assume �nally that Γ is a lattice in G and let F be a fundamental domain. Then F ×Uis a fundamental domain for Λ in G×D, which has �nite volume since a Haar measure forG ×D may be obtained by taking the product of respective Haar measures for G and D.Thus Λ has �nite covolume in G×D. �

Proof of Lemma 10.15. In order to construct a continuous homomorphism π : H → D, itsu�ces to check that any net in Λ whose image in H converges to the identity also convergesto the identity in D; this follows from the de�nitions of Γ and D since the net is eventuallyin any Λ-conjugate of U . Notice that π has dense image.

We claim that the kernel of π is⋂λ∈Λ λUλ

−1. Indeed, if on the one hand k ∈ ker(π)is the limit of the images in H of a net {λi} in Λ, then for any λ we have eventuallyλi ∈ λ−1Γλ ⊆ λ−1(G × U)λ so that indeed k ∈ λ−1Uλ since U is closed. Conversely, ifk ∈

⋂λ∈Λ λUλ

−1 is limit of images of {λi}, then, since U is open, for any λ the image of λiis eventually in λUλ−1, hence in λΓλ−1 so that π(λi)→ 1. This proves the claim.

Now it follows that ker(π) is indeed the core K of the statement since U is compact.The fact that π is onto and open follows from the existence of a compact open subgroup inH. �

Theorem 10.16. Let G be a locally compact group and Γ < G be a lattice. Assume that Gpossesses a �nitely generated dense subgroup Λ such that Γ < Λ < CommG(Γ).

If Λ admits a faithful Zariski-dense representation in an admissible semi-simple group(over any �eld), then the amenable radical R of G is compact and the quasi-centre QZ (G)is virtually contained in Γ · R. Furthermore, upon replacing G by a �nite index subgroup,the quotient G/R splits as G+ ×QZ (G/R) where G+ is a semi-simple algebraic group andthe image of Γ in G+ is an arithmetic lattice.

Proof. Let J = G×D, where D is the totally disconnected locally compact group providedby Lemma 10.14. As a totally disconnected group, it has numerous compact open subgroups(for instance the closure of Γ). We shall view Λ as an irreducible lattice in J . The projectionof Λ to G is faithful by construction. By Proposition 10.10, the pair (Λ, J) is superrigid.This allows us to apply Proposition 10.5. Since the amenable radical RG of G is containedin the amenable radical RJ of J , it is compact. Furthermore, the quasi-centre of G iscontained in the quasi-centre of J and the centre-free group G/RG is a direct factor ofJ+ ×QZ (J/RJ); the desired conclusions follow. �


10.E. Lattices in products of Lie and totally disconnected groups.

Theorem 10.17. Let Γ < G = S ×D be a �nitely generated irreducible lattice, where S isa connected semi-simple Lie group with trivial centre and D is a totally disconnected locallycompact group. Let ΓD �D be the canonical discrete kernel of D.

Then D/ΓD is a pro�nite extension of a semi-simple algebraic group Q and the image ofΓ in S ×Q, which is isomorphic to Γ/ΓD, is an arithmetic lattice.

Corollary 10.18. In particular, D is locally pro�nite by analytic. �

A family of examples will be constructed in Section 11.C below, showing that the state-ment cannot be simpli�ed even in a geometric setting (see Remark 11.7).

Proof of Theorem 10.17. By the very nature of the statement, we can factor out the canon-ical discrete kernel. Therefore, we shall assume henceforth that the projection map Γ→ Sis injective. We can also assume that S has no compact factors. Since S is connectedwith trivial centre, there is a Zariski connected semi-simple adjoint R-group H without R-anisotropic factors such that S = H(R). Notice that the injectivity of Γ → S is preservedwhen passing to �nite index subgroups.

By Proposition 10.10, the pair (Γ, G) is superrigid. We can therefore apply Proposi-tion 10.5. In particular, D has compact amenable radical and therefore, in view of thestatement of Theorem 10.17, we can assume that this radical is trivial. Given the conclu-sion of Proposition 10.5, it only remains to show that the quasi-centre QZ (G) of G is trivial.We now know that QZ (G) is virtually contained in Γ; since on the other hand S has trivialquasi-centre, QZ (G) ⊆ 1 ×D. In other words, QZ (G) is contained in the discrete kernelΓD, which has been rendered trivial. This completes the proof. �

We have treated Theorem 10.17 as a port of call on the way to Theorem 10.1. In fact,one can also describe lattices in products of groups with a simple algebraic factor over anarbitrary local �eld and in most cases without assuming �nite generation a priori. Werecord the following statement, which will not be used below.

Theorem 10.19. Let k be any local �eld and G an admissible connected absolutely almostsimple adjoint k-group. Let H be any compactly generated locally compact group admittinga compact open subgroup. Let Γ < G(k)×H be an irreducible lattice. In case k has positivecharacteristic and the k-rank of G is one, we assume Γ cocompact.

Then H/ΓH is a compact extension of a semi-simple algebraic group Q and the image ofΓ in G(k)×Q is an arithmetic lattice.

There is no assumption whatsoever on the compactly generated locally compact group Hbeyond admitting a compact open subgroup; recall that the latter is automatic ifH is totallydisconnected [Bou71, III �4 No 6]. Notice that a posteriori it follows from arithmeticitythat Γ is �nitely generated; in the proof below, �nite generation will be established in twosteps.

Proof of Theorem 10.19. We factor out the canonical discrete kernel ΓH and assume hence-forth that it is trivial. This does not a�ect the other assumptions and thus we choose somecompact open subgroup U < H. We write G = G(k) and consider ΓU = Γ ∩ (G × U)as in Lemma 8.2. Since we factored out the canonical discrete kernel, we can consider ΓUas a lattice in G commensurated by the dense subgroup Γ < G. Moreover, ΓU is �nitelygenerated; indeed, either we have simultaneously rankk(G) = 1 and char(k) > 0, in whichcase we assumed Γ cocompact, so that ΓU is cocompact in the compactly generated groupG(k) (again Lemma 8.2) and hence �nitely generated [Mar91, I.0.40]; or else, ΓU is known


to be �nitely generated by applying, as the case may be, either Kazhdan's property, orthe theory of fundamental domains, or the cocompactness of p-adic lattices � we refer toMargulis, Sections (3.1) and (3.2) of Chapter IX in [Mar91].

We can now apply Margulis' arthmeticity [Mar91, 1.(1)] and deduce that G is de�nedover a global �eld K and that ΓU is commensurable to G(K(S)) for some �nite set of placesS; in short ΓU is S-arithmetic. (The idea to obtain �rst this preliminary arithmeticity ofΓU was suggested by M. Burger.) It follows that Γ is rational over the global �eld K,see Theorem 3.b in [Bor66] (loc. cit. is formulated for the Lie group case; see [Wor07,Lemma 7.3] in general).

Since the pair (Γ, G × H) is superrigid (for instance by Proposition 10.10), only the apriori lack of �nite generation for Γ prevents us from applying Proposition 10.5. However, agood part of the proof of that proposition is already secured here since Γ has been shown tobe rational over a global �eld. We now proceed to explain how to adapt the remaining partof that proof to the current setting. We use those elements of notation introduced in theproof of Proposition 10.5 that do not con�ict with present notation and review all uses of�nite generation that are either explicit in the proof of Proposition 10.5 or implicit throughreferences to [Mon05].

The compact generation ofG×H is an assumption rather than a consequence of Lemma 7.12.We also used �nite generation in order to pass to a �nite index subgroup of Γ contained

in G(Kv)+ for all valuations v ∈ Σ. We shall postpone this step, so that the whole argu-mentation provides us with maps from G×H to a product Q of factors that lie in-betweenG(Kv)+ and G(Kv). In particular all these factors are quasi-simple and we can still appealto Corollary 5.15 as before. Notice however that at the very end of the proof, once �nitegeneration is granted, we can invoke the argument that G(Kv)/G(Kv)+ is virtually torsionAbelian [BT73, 6.14] and thus reduce again to the case where Γ is contained in G(Kv)+.

We now justify that the image of Γ in G×Q is discrete because previously this followedfrom [Mon05, (3.13)] which relies on �nite generation. If Γ were not discrete, an applicationof [BG07, Lemma 2.1] would provide a valuation v /∈ Σ with Γ unbounded in G(Kv), whichis absurd.

We are now in a situation where G×H maps to G×Q with cocompact �nite covolumeimage and injectively on Γ; therefore the discreteness of the image of Γ implies that thismap is proper and hence H is a compact extension of Q. Pushing forward the measure on(G×H)/Γ, we see that the image of Γ in G×Q is a lattice. Now Γ is �nitely generated (seeabove references to [Mar91, IX]) and thus the proof is completed as in Proposition 10.5.The discrete factor occurring in the conclusion of the latter proposition is trivial for thesame reason as in the proof of Theorem 10.17. �

10.F. Lattices in general products. We begin with the special case of totally discon-nected groups.

Proof of Theorem 10.6. An issue that we need to deal with is that the projection of Γ to G1

is a priori not faithful. In order to circumvent this di�culty, we proceed to a preliminaryconstruction.

Let ι : Γ→ Γ be the canonical map to the pro�nite completion of Γ and denote its kernelby Γ(f); in other words, Γ(f) is the �nite residual of Γ. Let G1 denote the locally compactgroup which is de�ned as the closure of the image of Γ in G1 × Γ under the product mapproj1×ι, where proj1 : G → G1 is the canonical projection. Since proj1(Γ) is dense in G1

and Γ is compact, the canonical map G1 → G1 is surjective. In other words, the group G1

is a compact extension of G1.


We now de�ne G′1 = G2×· · ·×Gn and G = G1×G′1. Then Γ admits a diagonal embeddinginto G through which the injection of Γ in G factors. We will henceforth identify Γ with itsimage in G and consider Γ as an irreducible lattice of G.

We claim that the pair (Γ, G) is superrigid.

The argument is a variation on the proof of Proposition 10.10. Let τ : Γ → H(k) beas in the de�nition of superrigid pairs. Since τ(Γ) is �nitely generated and linear, it isresidually �nite [Mal40]. This means that τ factors through Γ := Γ/Γ(f). Let U < G′1 bea compact open subgroup, ΓU = Γ ∩ (G1 × U) and ΓU = ΓU/(ΓU ∩ Γ(f)). By construction

and Lemma 8.2, we can consider ΓU as a lattice in G1 commensurated by Γ. Arguing asin Proposition 10.10, when τ(ΓU ) is unbounded one applies commensurator superrigidityyielding a continuous map J → H(k) and extending the map Γ → H(k) and hence also τ .When τ(ΓU ) is bounded, one applies Lemma 9.3 instead and the resulting extension factorsthrough G′1. This proves the claim.

In order to conclude that the pair (Γ, G) is also superrigid, it now su�ces to applyLemma 10.7. �

Corollary 10.20. Theorem 10.1 holds in the particular case of totally disconnected groups.

Proof. Theorem 10.6 provides the hypothesis needed for Proposition 10.5. �

We now turn to the general case Γ < G = G1 × · · · × Gn of Theorem 10.1. The mainpart of the remaining proof will consist of a careful analysis of how the lattice Γ might sitin various subproducts hidden in the factors Gi or their �nite index subgroups once theamenable radical has been trivialised. It will turn out that Γ is virtually a direct productΓ′ × Γ′′, where Γ′ is an irreducible lattice in a product S′ × D′ with S′ a semi-simple Lie(virtual) subproduct of G and D′ a totally disconnected subgroup of G whose position willbe clari�ed; as for Γ′′, it is an irreducible lattice in a semi-simple Lie group S′′ that turns outto satisfy the assumptions of Margulis' arithmeticity. Of course, any of the above factorsmight well be trivial.

Completion of the proof of Theorem 10.1. The amenable radical is compact by Corollary 10.13and hence we can assume that it is trivial. The group G (and hence also all �nite indexsubgroups and factors) is compactly generated by Lemma 7.12. Upon regrouping the lastn−1 factors and in view of the de�nition of an irreducible lattice (see p. 52), we can assumeG = G1 × G2. We apply the solution to Hilbert's �fth problem (compare Theorem 4.6)and write Gi = Si × Di after replacing G and Γ with �nite index subgroups. Here Siare connected semi-simple centre-free Lie groups without compact factors and Di totallydisconnected compactly generated with trivial amenable radical. Set S = S1 × S2 andD = D1 × D2. Thus Γ is a lattice in G = S × D. Notice that if S is trivial, then G istotally disconnected and we are done by Theorem 10.6. We assume henceforth that S isnon-trivial. The main remaining obstacle is that the lattice Γ need not be irreducible withrespect to the product decomposition G = S ×D.

Observe that the closure projD(Γ) of the projection of Γ to D has trivial amenable radical.

Indeed projDi(Γ) is dense in Γi for i = 1, 2, hence the projection projD(Γ) → Di hasdense image. The desired claim follows since G, and hence Di, has trivial amenable radical.

Let U < D be a compact open subgroup and set ΓU = Γ ∩ (S × U). By Lemma 8.2,the projection projS(ΓU ) of ΓU to S is a lattice which is commensurated by projS(Γ). Thelattice projS(ΓU ) possesses a �nite index subgroup which admits a canonical splitting into�nitely many irreducible groups Γ1×· · ·×Γr, compare Theorem 8.17. Furthermore each Γi


is an irreducible lattice in a semi-simple subgroup Si < S which is obtained by regroupingsome of the simple factors of S.

Since the projection of Γ to each G1 and G2, and hence to S1 and S2, is dense, it followsthat the projection of Γ to each simple factor of S is dense. We now consider the projectionof Γ to the various factors Si. In view of the preceding remark and the fact that Γi isan irreducible lattice in Si, it follows that projSi(Γ) is either dense in Si or discrete andcontains Γi with �nite index, see [Mar91, IX.2.7]. Let now

S′ = 〈Si | projSi(Γ) is non-discrete〉 and S′′ = 〈Si | projSi(Γ) is discrete〉.

We claim that the projection of Γ to S′ is dense.

If this failed, then by [Mar91, IX.2.7] there would be a subproduct of some simple factorsof S′ on which the projection of Γ is a lattice. Since each Γi is irreducible, this subproductis a regrouping

Si1 × · · · × Sip

of some factors Si. Now the projection of Γ is a lattice in this subgroup, hence it containsthe product Γi1 × · · · × Γip with �nite index and thus projects discretely to each Sij . Thiscontradicts the de�nition of S′ and proves the claim.

Our next claim is that Γ has a �nite index subgroup which splits as Γ′ × Γ′′, where Γ′′ =projS′′(Γ) and Γ′ is a lattice in S′ ×D.

In order to establish this, we de�ne

Γ′ = Ker(proj : Γ→ S′′) and Γ′′ =⋂γ∈Γ

γΓUγ−1.

Notice that Γ′ and Γ′′ are both normal subgroups of Γ. Since projD(Γ′′) is a compactsubgroup of D normalised by projD(Γ), which has trivial amenable radical, it follows thatΓ′′ ⊂ S′×S′′×1. Therefore, the intersection Γ′∩Γ′′ is a normal subgroup of Γ contained inS′ × 1× 1. In view of the preceding claim, we deduce that Γ′ ∩ Γ′′ = 1. Thus 〈Γ′ ∪ Γ′′〉� Γis isomorphic to Γ′ × Γ′′. Since Γ′U = Γ′ ∩ ΓU projects to a lattice in S′ which commuteswith the projection of Γ′′, we deduce moreover that projS′(Γ′′) = 1, or equivalently thatΓ′′ < 1× S′′ × 1.

Since the projection of Γ to S′′ has discrete image by de�nition, it follows from Propo-sition 8.1 that Γ′ < S′ × 1 ×D projects onto a lattice in S′ ×D. On the other hand, thevery de�nition of S′′ implies projS′′(Γ) contains projS′′(ΓU ), and hence also projS′′(Γ′′), asa �nite index subgroup. In particular, this shows that Γ′ × Γ′′ is a lattice in S′ × S′′ ×D.Since it is contained in the lattice Γ, we �nally deduce that the index of Γ′ × Γ′′ in Γ is�nite.

We observe that we have in particular obtained a lattice Γ′′ < S′′ with S′′ non-simpleand Γ′′ irreducible (unless both Γ′′ and S′′ are trivial), because the projection of Γ to anysimple Lie group factor is dense: indeed, any simple factor must be a factor of some Giand Γ projects densely on Gi. It follows from Margulis' arithmeticity theorem [Mar91,Theorem 1.(1')] that Γ′′ is an arithmetic lattice in S′′.

Turning to the other lattice, we remark that Γ′ admits a faithful Zariski-dense represen-tation in a semi-simple group, obtained by reducing the given representation of Γ. Further-more, notice that the projection of Γ to S′ coincides (virtually) with the projection of Γ. Inparticular it has dense image. Therefore, setting D′ = projD(Γ′), we may now view Γ′ asan irreducible lattice in S′ ×D′. We may thus apply Theorem 10.17. Notice that the sameargument as before shows that D′ has trivial amenable radical.


We claim that the canonical discrete kernel Γ′D′ is in fact a direct factor of D′.

Indeed, since Γ is residually �nite by Malcev's theorem [Mal40], Proposition 8.24 ensuresthat Γ′D′ centralises the discrete residual D′(∞). In particular D′(∞) ∩ Γ′D′ = 1 since D′

has trivial amenable radical. Furthermore, since D′/Γ′D′ is a semi-simple group, its discreteresidual has �nite index. In particular, upon replacing D′ by a �nite index subgroup wehave D′ ∼= D′(∞) × Γ′D′ as desired. It also follows that Γ′D′ itself admits a Zariski-denserepresentation in a semi-simple group.

It remains to consider again the projection maps projDi : D → Di. Restricting these mapsto D′ and using the fact that projDi(D

′) is dense, we obtain that Di∼= projDi(D

′(∞))×D′i,where D′i = projDi(Γ

′D′). The �nal conclusion follows by applying Corollary 10.20 to the

irreducible lattice Γ′D′ < D′1 ×D′2. �

Proof of Corollary 10.3. Since Γ is �nitely generated and irreducible, all Gi are compactlygenerated (alternatively, apply Lemma 7.12).

We claim that all projections Γ→ Gi are injective. Indeed, if not, then (by induction onn) there is j such that the canonical discrete kernel ΓGj is non-trivial. It is then cocompact,which implies that the projection

Gj/ΓGj ×∏i 6=j

Gi −→∏i 6=j

Gi

is proper. This is contradicts the fact that the image of Γ in the left hand side above isdiscrete whilst it is dense in the right hand side, proving the claim.

Suppose given a linear representation of Γ in characteristic 6= 2, 3 whose image is notvirtually soluble. Arguing as in [Mon05], we can reduce to the case where we have a Zariski-dense representation τ : Γ → H(K) in a non-trivial connected adjoint absolutely simplegroup H over a �nitely generated �eld K. Since τ(Γ) is in�nite, we can choose a completionk of K for which τ(Γ) is unbounded [BG07, 2.1].

Part of the argument in [Mon05] is devoted to proving that the representation is a pos-teriori faithful. One can adapt the entire proof to the present setting, but we propose analternative line of reasoning using an amenability theorem from [BS06]. Suppose towardsa contradiction that the kernel Γ0 � Γ of τ is non-trivial. Since the projections are in-jective, the closure Ni of the image of Γ0 in Gi is a non-trivial closed subgroup, which isnormal by irreducibility and hence is cocompact. Then Theorem 1.3 in [BS06] implies thatΓ/Γ0 is amenable, contradicting the fact that τ(Γ) is not virtually soluble in view of Tits'alternative [Tit72].

At this point we can conclude by Theorem 10.1. �

11. Geometric arithmeticity

11.A. CAT(0) lattices and parabolic isometries. We now specialise the various arith-meticity results of Section 10 to the case of lattices in CAT(0) spaces and combine themwith some of our geometric results.

Recall that a parabolic isometry is called neutral if it has zero translation length; thefollowing contains Theorem 1.25 from the Introduction.

Theorem 11.1. Let X be a proper CAT(0) space with cocompact isometry group and Γ <G := Is(X) be a �nitely generated lattice. Assume that Γ is irreducible and that G containsa neutral parabolic isometry. Then one of the following assertions holds:

(i) G is a non-compact simple Lie group of rank one with trivial centre.


(ii) There is a subgroup ΓD ⊆ Γ normalised by G, which is either �nite or in�nitelygenerated and such that the quotient Γ/ΓD is an arithmetic lattice in a product ofsemi-simple Lie and algebraic groups.

Proof. Let X ′ ⊆ X be the canonical subspace provided by Theorem 8.11; notice that X ′

still admits a neutral parabolic isometry. Theorem 1.1 and its addendum now apply to X ′.The space X ′ has no Euclidean factor: indeed, otherwise Theorem 8.8 would imply X ′ = R,which has no parabolic isometries. The kernel of the Γ-action on X ′ is �nite and we willinclude it in the subgroup ΓD below.

We distinguish two cases according as X ′ has one or more factors.In the �rst case, Is(X ′) cannot be totally disconnected since otherwise Corollary 5.3(i)

rules out neutral parabolic isometries. Thus Is(X ′) is a non-compact simple Lie group withtrivial centre. If its real rank is one, we are in case (i); otherwise, Γ is arithmetic by Margulis'arithmeticity theorem [Mar91, Theorem 1.(1')] and we are in case (ii).

For the rest of the proof we treat the case of several factors for X ′; let Γ∗ and let HΓ∗

be as in Section 8.E. Note that HΓ∗ acts cocompactly on each irreducible factor of X ′.Furthermore, each irreducible factor of HΓ∗ is non-discrete by Theorem 8.17. ThereforeHΓ∗ is a product of the form S × D (possibly with one trivial factor), where S is a semi-simple Lie group with trivial centre and D is a compactly generated totally disconnectedgroup without discrete factor.

By Corollary 5.3(i), the existence of a neutral parabolic isometry in G implies that Is(X ′)is not totally disconnected. Lemma 8.23 ensures that the identity component of Is(X ′) isin fact contained in HΓ∗ . Therefore, upon passing to a �nite index subgroup, the identitycomponent of Is(X ′) coincides with S.

If D is trivial, then HΓ∗ = S is a connected semi-simple Lie group containing Γ as anirreducible lattice. Since S is non-simple, it has higher rank and we may appeal again toMargulis' arithmeticity theorem; thus we are done in this case.

Otherwise, D is non-trivial and we may then apply Theorem 10.17. It remains to checkthat the normal subgroup ΓD < Γ, if non-trivial, is not �nitely generated. But we knowthat ΓD is a discrete normal subgroup of D. By Theorem 7.4, the lattice Γ, and hencealso HΓ∗ , acts minimally without �xed point at in�nity on each irreducible factor of X ′.Therefore, Corollary 4.8 ensures that D has no �nitely generated discrete normal subgroup,as desired. �

Here is another variation, of a more geometric �avour; this time, it is not required thatthere be a neutral parabolic isometry:

Theorem 11.2. Let X be a proper geodesically complete CAT(0) space with cocompactisometry group and Γ < Is(X) be a �nitely generated lattice. Assume that Γ is irreducibleand residually �nite.

If G := Is(X) contains any parabolic isometry, then X is a product of symmetric spacesand Bruhat�Tits buildings. In particular, Γ is an arithmetic lattice unless X is a real orcomplex hyperbolic space.

Proof. We maintain the notation of the previous proof and follow the same arguments.We do not know a priori whether there exists a neutral parabolic isometry. However,under the present assumption that X is geodesically complete, Corollary 5.3(iii) showsthat the existence of any parabolic isometry is enough to ensure that Is(X ′) is not totallydisconnected. Thus the conclusion of Theorem 11.1 holds. In case (i), Theorem 6.4(iii)ensures that X is a rank one symmetric space and we are done. We now assume that (ii)holds and de�ne D as in the proof of Theorem 11.1.


The canonical discrete kernel ΓD is trivial by Theorem 8.26. Since D has no non-trivialcompact normal subgroup by Corollary 4.8, it follows from Theorem 10.17 thatD is a totallydisconnected semi-simple algebraic group. Therefore, the desired result is a consequence ofTheorem 6.4(iii). �

For the record, we propose a variant of Theorem 11.2:

Theorem 11.3. Let X be a proper geodesically complete CAT(0) space with cocompactisometry group and Γ < Is(X) be a �nitely generated lattice. Assume that Γ is irreducibleand that every normal subgroup of Γ is �nitely generated.

If G := Is(X) contains any parabolic isometry, then X is a product of symmetric spacesand Bruhat�Tits buildings of total rank ≥ 2. In particular, Γ is an arithmetic lattice.

Proof. As for Theorem 11.2, we can apply Theorem 11.1. We claim that case (i) is ruledout under the current assumptions. Indeed, a lattice in a simple Lie group of rank one isrelatively hyperbolic (see [Far98] or [Osi06]) and as such has numerous in�nitely generatednormal subgroups (and is even SQ-universal, see [Gro87] or [Del96] for the hyperbolic caseand [AMO07] for the general relative case). In case (ii) the discrete kernel ΓD is trivial andrank one is excluded as in case (i) if the group is Archimedean; if it is non-Archimedean,then there are no non-uniform �nitely generated lattices (see [BL01]) and thus Γ is againGromov-hyperbolic which contradicts the assumption on normal, subgroups as before. �

We can now complete the proof of some results stated in the Introduction.

Proof of Theorem 1.23. If Γ is residually �nite, then Theorem 11.2 yields the desired conclu-sion; it therefore remains to consider the case where Γ is not residually �nite. We follow thebeginning of the proof of Theorem 11.2 until the invocation of Theorem 8.26, since the latterno longer applies. However, we still know that there is a non-trivial Lie factor in Is(X ′) andtherefore we apply Theorem 10.17 in order to obtain the desired conclusion about the latticeΓ. As for the symmetric space factor of the space, it is provided by Theorem 6.4(iii). �

Proof of Corollary 1.24. One implication is given by Theorem 1.23. For the converse, itsu�ces to recall that unipotent elements exist in all semi-simple Lie groups of positive realrank. �

11.B. Arithmeticity of linear CAT(0) lattices. We start by considering CAT(0) latticeswith a linear non-discrete linear commensurator:

Theorem 11.4. Let X be a proper geodesically complete CAT(0) space with cocompactisometry group and Γ < Is(X) be a �nitely generated lattice. Assume that Is(X) possessesa �nitely generated subgroup Λ containing Γ as a subgroup of in�nite index, and commen-surating Γ.

If X is irreducible and Λ possesses a faithful �nite-dimensional linear representation (incharacteristic 6= 2, 3), then X is a symmetric space or a Bruhat�Tits building; in particularΓ is an arithmetic lattice.

Remark 11.5. Several examples of irreducible CAT(0) spaces X of dimension > 1 ad-mitting a discrete cocompact group of isometries with a non-discrete commensurator inIs(X) have been constructed by F. Haglund [Hag98] and A. Thomas [Tho06] (see also [Hag,Théorème A] and [BT]). In all cases that space X is endowed with walls; in particular Xis the union of two proper closed convex subspaces. This implies in particular that X isnot a Euclidean building. Therefore, Theorem 11.4 has the following consequence: in theaforementioned examples of Haglund and Thomas, either the commensurator of the lattice


is nonlinear, or it is the union of a tower of lattices. In fact, as communicated to us byF. Haglund, for most of these lattices the commensurator contains elliptic elements of in-�nite order; this implies right away that the commensurator is not an ascending union oflattices and, hence, it is nonlinear. Note on the other hand that it is already known thatIs(X) is mostly nonlinear in these examples, since it contains closed subgroups isomorphicto the full automorphism group of regular trees.

Proof of Theorem 11.4. Since X is irreducible and the case X = R satis�es the conclusionsof the theorem, we assume henceforth that X has no Euclidean factor.

The Is(X)-action on X is minimal by Lemma 2.13 and has no �xed point at in�nity byCorollary 8.12. In particular, we can apply Theorem 1.11: either Is(X) is totally discon-nected or it is simple Lie group with trivial centre and X is the associated symmetric space.In the latter case, Margulis' arithmeticity theorem �nishes the proof. We assume henceforththat Is(X) is totally disconnected.

Let G denote the closure of Λ in Is(X). Note that G acts minimally without �xed pointat in�nity, since it contains a subgroup, namely Γ, which possesses these properties byTheorem 7.4. In particular G has trivial amenable radical by Theorem 1.6 and thus thesame holds for the dense subgroup Λ < G. In particular any faithful representation of Λ toan algebraic group yields a faithful representation of Λ to an adjoint semi-simple algebraicgroup with Zariski-dense image, to which we can apply Theorem 10.16. As we saw, thegroup G has no non-trivial compact (in fact amenable) normal subgroup and furthermoreG is irreducible since X is so, see Theorem 1.6. The fact that the lattice Γ has in�niteindex in Λ rules out the discrete case. Therefore G is a simple algebraic group and Γ anarithmetic lattice.

It remains to deduce that X has the desired geometric shape. This will follow fromTheorem 6.4(iii) provided we show that ∂X is �nite-dimensional and that G has full limitset. The �rst fact holds since X is cocompact; the second is provided by Corollary 7.10. �

Remark 11.5 illustrates that Theorem 11.4 fails dramatically if one assumes only that Γis linear. However, passing now to the case where X is reducible, the linearity of Γ is enoughto establish arithmeticity, independently of any assumption on commensurators, the resultannounced in Theorem 1.26 in the Introduction.

Theorem 11.6. Let X be a proper geodesically complete CAT(0) space with cocompactisometry group and Γ < Is(X) be a �nitely generated lattice. Assume that Γ is irreducibleand possesses some faithful linear representation (in characteristic 6= 2, 3).

If X is reducible, then Γ is an arithmetic lattice and X is a product of symmetric spacesand Bruhat�Tits buildings.

Proof. In view of Theorem 8.8, we can assume that X has no Euclidean factor. The Is(X)-action on X is minimal by Lemma 2.13 and has no �xed point at in�nity by Corollary 8.12.In particular, we can apply Theorem 1.11 to obtain decompositions of Is(X) and X in whichthe factors of X corresponding to connected factors of Is(X) are isometric to symmetricspaces. There is no loss of generality in assuming Γ∗ = Γ in the notation of Section 8.E.Let now G be the hull of Γ. By Remark 8.21, the group Γ is an irreducible lattice in G.

Since Is(X) acts minimally without �xed point at in�nity, it follows from Corollary 7.7that Γ has trivial amenable radical. In particular any faithful representation of Γ to analgebraic group yields a faithful representation of Γ to an adjoint semi-simple algebraicgroup with Zariski-dense image, to which we can apply Theorem 10.1.


The group G has no non-trivial compact normal subgroup e.g. by minimality. Further-more the discrete factor is trivial by Theorem 8.17. Therefore G is a simple algebraic groupand Γ an arithmetic lattice.

It remains to deduce that X has the desired geometric shape and this follows exactly asin the proof of Theorem 11.4. �

11.C. A family of examples. We shall now construct a family of lattices Γ < G = S×Das in the statement of Theorem 10.17 (see also Theorem 11.2) with the following additionalproperties:

(i) There is a proper CAT(0) space Y with D < Is(Y ) such that the D-action iscocompact, minimal and without �xed point at in�nity. In particular, setting X =XS × Y , where XS denotes the symmetric space associated to S, the Γ-action onX is properly discontinuous (in fact free), cocompact, minimal, without �xed pointat in�nity.

(ii) The canonical discrete kernel ΓD�D is in�nite (in fact, it is a free group of countablerank).

(iii) The pro�nite kernel of D/ΓD → Q is non-trivial.

Remark 11.7. Since D is minimal, it has no compact normal subgroup and thus we seethat the pro�nite extension appearing in Theorem 10.17 cannot be eliminated.

We begin with a general construction:

Let g be (the geometric realisation of) a locally �nite graph (not reduced to a single point)and let Q < Is(g) a closed subgroup whose action is vertex-transitive. In particular, Q isa compactly generated totally disconnected locally compact group. We point out that anycompactly generated totally disconnected locally compact group can be realised as acting onsuch a graph by considering Schreier graphs g, see [Mon01, �11.3]; the kernel of this actionis compact and arbitrary small. On the other hand, if Q is a non-Archimedean semi-simplegroup, one can also take very explicit graphs drawn on the Bruhat�Tits building of Q, e.g.the 1-skeleton (this part is inspired by [BM00a, 1.8], see also [BMZ04]).

Let moreover C be an in�nite pro�nite group and choose a locally �nite rooted tree t witha level-transitive C-action for which every in�nite ray has trivial stabiliser. For instance,one can choose the coset tree associated to any nested sequence of open subgroups withtrivial intersection, see the proof of Théorème 15 in �6 on p. 82 in [Ser77]. We de�ne alocally �nite graph h with a C × Q-action as the 1-skeleton of the square complex t × g.Let a = h be the universal cover of h, Λ = π1(h) and de�ne the totally disconnected locallycompact group D by the corresponding extension

1 −→ Λ −→ D −→ C ×Q −→ 1.

Proposition 11.8. There exists a proper CAT(0) space Y such that D sits in Is(Y ) as aclosed subgroup whose action is cocompact, minimal and without �xed point at in�nity.

Proof. One veri�es readily the following:

Lemma 11.9. Let a be (the geometric realisation of) a locally �nite simplicial tree andD < Is(a) any subgroup. Let x ∈ a be a vertex and let Y be the completion of the metricspace obtained by assigning to each edge of a the length 2−r, where r is the combinatorialdistance from this edge to the nearest point of the orbit D.x.


Then Y is a proper CAT(0) space with a cocompact continuous isometric D-action. More-over, if the D-action on a was minimal or without �xed point at in�nity, then the corre-sponding statement holds for the D-action on Y . �

Apply the lemma to the tree a = h considered earlier. We claim that the D-action ona is minimal. Clearly it su�ces to show that the Λ-action is minimal. Note that Λ actstransitively on each �bre of p : h → h. Thus it is enough to show that the convex hull ofa given �bre meets every other �bre. Consider two distinct vertices v, v′ ∈ h. The productnature of h makes it clear that v and v′ are both contained in a common minimal loopbased at v. This loop lifts to a geodesic line in h which meets the respective �bres of v andv′ alternatively and periodically. In particular, this construction yields a geodesic segmentjoining two points in the �bre above v and containing a point sitting above v′, whence theclaim.

Since Λ acts freely and minimally on the tree a which is not reduced to a line, it followsthat Λ �xes no end of a. Thus the lemma provides a proper CAT(0) space Y with acocompact minimal isometric D-action, without �xed point at in�nity. It remains to showthat D < Is(Y ) is closed. This holds because the totally disconnected groups Is(a) andIs(Y ) are isomorphic; indeed, the canonical map a → Y induces a continuous surjectivehomomorphism Is(a)→ Is(Y ), which is thus open. �

Remark 11.10. The above construction gives an example of a proper CAT(0) space witha totally disconnected cocompact and minimal group of isometries such that not all pointstabilisers are open. Consider indeed the points added when completing. Their stabilisersmap to Q under D → (C × Q) and hence cannot be open. In other words, the action isnot smooth in the terminology of [Cap07]. Notice however that the set of points with openstabiliser is necessarily a dense convex invariant set.

We shall now specialise this general construction to yield our family of examples. LetK, H, Σ, K(Σ) be as described after Theorem 10.1 on p. 60. We write Σf ,Σ∞ ⊆ Σfor the subsets of �nite/in�nite places and assume that both are non-empty. Let S =∏v∈Σf

H(Kv)+ and Q =∏v∈Σ∞

H(Kv)+. The group ∆ = H(K(Σ)) ∩ (S × Q) is anirreducible cocompact lattice in S×Q. Let C be any pro�nite group with a dense inclusion∆ → C. We now embed ∆ diagonally in S × C × Q; clearly ∆ is a cocompact lattice.Let Γ < G = S ×D be its pre-image. Then Γ is a cocompact lattice since it contains thediscrete kernel of the canonical map G→ S ×C ×Q. It is clearly irreducible and thereforeprovides an example that the structure of the description in the conclusion of Theorem 10.17cannot be simpli�ed. Furthermore, the normal subgroup appearing in Theorem 11.1(ii) isalso unavoidable.

We end this section with a few supplementary remarks on the preceding construction:

(i) If the pro�nite group C has no discrete normal subgroup, then ΓD = π1(h) coincideswith the quasi-centre of D. This would be the case for example if C = H(Kv) andH is almost K-simple of higher rank, where v is a non-Archimedean valuation suchthat H is Kv-anisotropic. In particular, in that situation ΓD is the unique maximaldiscrete normal subgroup of D and the quotient D/ΓD has a unique maximalcompact normal subgroup. Thus the group G admits a unique decomposition as inTheorem 10.17 in this case.

(ii) We emphasise that, even though D/ΓD decomposes as a direct product C × Qin the above construction, the group D admits no non-trivial direct product de-composition, since it acts minimally without �xed point at in�nity on a tree (seeTheorem 1.6).


(iii) The fact that D/ΓD decomposes as a direct product C × Q is not a coincidence.In fact, this is happens always provided that every cocompact lattice in S has theCongruence Subgroup Property (CSP). Indeed, given a compact open subgroup Uof D/ΓD, the intersection ΓU of Γ ∩ (S × U) is an irreducible lattice in S × Uwith trivial canonical discrete kernels. By (CSP), upon replacing ΓU by a �niteindex subgroup (which amounts to replace U by an open subgroup), the pro�nite

completion ΓU splits as the product over all primes p of the pro-p completions (ΓU )p,

which are just-in�nite. Thus the canonical surjective map ΓU → U shows that Uis a direct product. This implies that the maximal compact normal subgroup ofD/ΓD is a direct factor.

(iv) According to a conjecture of Serre's (footnote on page 489 in [Ser70]), if S hashigher rank then every irreducible lattice in S has (CSP). (See [Rag04] for a recentsurvey on this conjecture.)

12. Some open questions

We conclude by collecting some perhaps noteworthy questions that we have encounteredwhile working on this paper, but that we have not been able to answer.

It is well known that the Tits boundary of a proper CAT(0) space with cocompactisometry group is necessarily �nite-dimensional (see [Kle99, Theorem C]). It is quite possiblethat the same conclusion holds under a much weaker assumption.

Question 12.1. Let X be a proper CAT(0) space such that Is(X) has full limit set. Is theboundary ∂X �nite-dimensional?

A positive answer to this question would show in particular that the second set of as-sumptions � denoted (b) � in Theorem 6.4 is in fact redundant.

Let G be a simple Lie group acting continuously by isometries on a proper CAT(0) spaceX. The Karpelevich�Mostow theorem ensures that there exists a convex orbit when X is asymmetric space of non-compact type. This statement, however, cannot be generalised toarbitrary X in view of Example 6.7.

Question 12.2. Let G be a simple Lie group acting continuously by isometries on a properCAT(0) space X. If the action is cocompact, does there exist a convex orbit?

It is shown in Theorem 6.4(iii) that if X is geodesically complete, then the answer ispositive. It is good to keep in mind Example 6.6, which shows that the natural analogueof this question for a simple algebraic group over a non-Archimedean local �eld has anegative answer. More optimistically, one can ask for a convex orbit whenever the simple Liegroup acts on a complete (not necessarily proper) CAT(0) space, but assuming the actionnon-evanescent (in the sense of [Mon06]). A positive answer would imply superrigiditystatements upon applying it to spaces of equivariant maps.

Many of our statements on CAT(0) lattices require the assumption of �nite generation.One should of course wonder for each of them whether it remains valid without this as-sumption. One instance where this question is especially striking is the following (seeCorollary 8.12).

Question 12.3. LetX be a proper CAT(0) space which is minimal and cocompact. Assumethat Is(X) contains a lattice. Is it true that Is(X) has no �xed point at in�nity?


Notice that Is(X) is compactly generated (see Lemma 8.3). In particular, if Is(X) is dis-crete, then it is �nitely generated and is itself a lattice. Therefore, in view of Corollary 8.12,one can assume that Is(X) is non-discrete in Question 12.3.

We have seen in Corollary 6.12 that if the isometry group of a proper CAT(0) spaceX is non-discrete in a strong sense, then Is(X) comes close to being a direct product oftopologically simple groups.

Question 12.4. Retain the assumptions of Corollary 6.12. Is it true that soc(G∗) is a prod-uct of simple groups? Is it cocompact in G, or at least does G have compact Abelianisation?

Clearly Corollary 6.12 reduces the question to the case where Is(X) is totally discon-nected. One can also ask if soc(G∗) is compactly generated (which is the case e.g if it iscocompact in G). If so, we obtain additional information by applying Proposition 5.11.

In the above situation one furthermore expects that the geometry of X is encoded in thestructure of Is(X). In precise terms, we propose the following.

Question 12.5. Retain the assumptions of Corollary 6.12. It it true that any propercocompact action ofG on a proper CAT(0) space Y yields an equivariant isometry ∂X → ∂Ybetween the Tits boundaries? Or an equivariant homeomorphism between the boundarieswith respect to the cône topology?

The discussion around Corollary 10.3 (cf. Remark 10.4) suggests the following.

Question 12.6. Let Γ < G = G1 × · · · × Gn be an irreducible �nitely generated lattice,where each Gi is a locally compact group. Does every character Γ→ R extend continuouslyto G?

Y. Shalom [Sha00] proved that this is the case when Γ is cocompact and in some othersituations.

References

[AB94] Norbert A'Campo and Marc Burger, Réseaux arithmétiques et commensurateur d'après G. A.Margulis, Invent. Math. 116 (1994), no. 1-3, 1�25.

[AB98a] Scot Adams and Werner Ballmann, Amenable isometry groups of Hadamard spaces, Math. Ann.312 (1998), no. 1, 183�195.

[AB98b] Stephanie B. Alexander and Richard L. Bishop, Warped products of Hadamard spaces,Manuscripta Math. 96 (1998), no. 4, 487�505.

[AB04] , Curvature bounds for warped products of metric spaces, Geom. Funct. Anal. 14 (2004),no. 6, 1143�1181.

[Ale57] Alexander D. Alexandrow, Über eine Verallgemeinerung der Riemannschen Geometrie, Schr.Forschungsinst. Math. 1 (1957), 33�84.

[AMO07] Goulnara Arzhantseva, Ashot Minasyan, and Denis Osin, The SQ-universality and residual prop-erties of relatively hyperbolic groups, J. Algebra 315 (2007), no. 1, 165�177.

[Bal95] Werner Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar, vol. 25, BirkhäuserVerlag, Basel, 1995, With an appendix by Misha Brin.

[Ber] Valeri�� N. Berestovski��, Busemann spaces of Aleksandrov curvature bounded above, Max-Planckpreprint Nr. 74/2001.

[Ber02] Valeri�� N. Berestovski��, Busemann spaces with upper-bounded Aleksandrov curvature, Algebra iAnaliz 14 (2002), no. 5, 3�18.

[BEW08] Yiftach Barnea, Mikhail Ershov, and Thomas Weigel, Abstract commensurators of pro�nitegroups, Preprint, 2008.

[BG02] Nadia Benakli and Yair Glasner, Automorphism groups of trees acting locally with a�ne permu-tations, Geom. Dedicata 89 (2002), 1�24.


[BG07] Emmanuel Breuillard and Tsachik Gelander, A topological Tits alternative, Ann. of Math. (2) 166(2007), no. 2, 427�474.

[BGM04] Janez Bernik, Robert Guralnick, and Mitja Mastnak, Reduction theorems for groups of matrices,Linear Algebra Appl. 383 (2004), 119�126.

[BGP92] Yuri Burago, Mikhaïl Gromov, and Gregory Perel′man, A. D. Aleksandrov spaces with curvaturesbounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3�51, 222.

[BGS85] Werner Ballmann, Mikhaïl Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature,Progress in Mathematics, vol. 61, Birkhäuser Boston Inc., Boston, MA, 1985.

[BH99] Martin R. Bridson and André Hae�iger, Metric spaces of non-positive curvature, Grundlehren derMathematischen Wissenschaften 319, Springer, Berlin, 1999.

[BL01] Hyman Bass and Alexander Lubotzky, Tree lattices, Progress in Mathematics, vol. 176, BirkhäuserBoston Inc., Boston, MA, 2001, With appendices by Bass, L. Carbone, Lubotzky, G. Rosenbergand J. Tits.

[BL05] Andreas Balser and Alexander Lytchak, Centers of convex subsets of buildings, Ann. Global Anal.Geom. 28 (2005), no. 2, 201�209.

[BM96] Marc Burger and Shahar Mozes, CAT(-1)-spaces, divergence groups and their commensurators,J. Amer. Math. Soc. 9 (1996), 57�93.

[BM00a] Marc Burger and Shahar Mozes, Groups acting on trees: from local to global structure, Inst.Hautes Études Sci. Publ. Math. (2000), no. 92, 113�150 (2001).

[BM00b] , Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. (2000), no. 92, 151�194(2001).

[BM02] Marc Burger and Nicolas Monod, Continuous bounded cohomology and applications to rigiditytheory, Geom. Funct. Anal. 12 (2002), no. 2, 219�280.

[BMZ04] Marc Burger, Shahar Mozes, and Robert J. Zimmer, Linear representations and arithmeticity forlattices in products of trees, FIM preprint, 2004.

[BO69] Richard L. Bishop and Barrett O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc.145 (1969), 1�49.

[Bon04] Patrick Bonvin, Strong boundaries and commensurator super-rigidity, Geom. Funct. Anal. 14(2004), no. 4, 843�848, Appendix to Topological simplicity, commensurator super-rigidity andnon-linearities of Kac-Moody groups by B. Rémy.

[Bor60] Armand Borel, Density properties for certain subgroups of semi-simple groups without compactcomponents, Ann. of Math. (2) 72 (1960), 179�188.

[Bor66] , Density and maximality of arithmetic subgroups, J. Reine Angew. Math. 224 (1966),78�89.

[Bou60] Nicolas Bourbaki, Éléments de mathématique. Première partie. (Fascicule III.) Livre III; Topolo-gie générale. Chap. 3: Groupes topologiques. Chap. 4: Nombres réels, Troisième édition revue etaugmentée, Actualités Sci. Indust., No. 1143. Hermann, Paris, 1960.

[Bou68] , Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV:Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des ré�exions.Chapitre VI: systèmes de racines, Actualités Scienti�ques et Industrielles, No. 1337, Hermann,Paris, 1968.

[Bou71] , Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971.[BS87] Marc Burger and Viktor Schroeder, Amenable groups and stabilizers of measures on the boundary

of a Hadamard manifold, Math. Ann. 276 (1987), no. 3, 505�514.[BS06] Uri Bader and Yehuda Shalom, Factor and normal subgroup theorems for lattices in products of

groups, Invent. Math. 163 (2006), no. 2, 415�454.[BT] Angela Barnhill and Anne Thomas, Density of commensurators for right-angled buildings,

Preprint.[BT73] Armand Borel and Jacques Tits, Homomorphismes �abstraits� de groupes algébriques simples,

Ann. of Math. (2) 97 (1973), 499�571.[Bur95] Marc Burger, Rigidity properties of group actions on CAT(0)-spaces, Proceedings of the Interna-

tional Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, pp. 761�769.[Cap07] Pierre-Emmanuel Caprace, Amenable groups and hadamard spaces with a totally disconnected

isometry group, Preprint, available at arXiv:0705.1980v1, 2007.[CFP96] James W. Cannon, William J. Floyd, and Walter R. Parry, Introductory notes on Richard Thomp-

son's groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215�256.


[Che99] Chien-Hsiung Chen, Warped products of metric spaces of curvature bounded from above, Trans.Amer. Math. Soc. 351 (1999), no. 12, 4727�4740.

[CK00] Christopher B. Croke and Bruce Kleiner, Spaces with nonpositive curvature and their ideal bound-aries, Topology 39 (2000), no. 3, 549�556.

[CL] Pierre-Emmanuel Caprace and Alexander Lytchak, In preparation.[CM08] Pierre-Emmanuel Caprace and Nicolas Monod, Some properties of non-positively curved lattices,

C. R. Math. Acad. Sci. Paris 0 (2008), no. 0, 0000�0000.[Del96] Thomas Delzant, Sous-groupes distingués et quotients des groupes hyperboliques., Duke Math. J.

83 (1996), no. 3, 661�682.[dR52] Georges de Rham, Sur la reductibilité d'un espace de Riemann, Comment. Math. Helv. 26 (1952),

328�344.[Ebe80] Patrick Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math. (2) 111 (1980), no. 3,

435�476.[Ebe82] , Isometry groups of simply connected manifolds of nonpositive curvature II, Acta Math.

149 (1982), no. 1-2, 41�69.[Ebe83] , Euclidean de Rham factor of a lattice of nonpositive curvature, J. Di�erential Geom. 18

(1983), no. 2, 209�220.[Ebe96] , Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University

of Chicago Press, Chicago, IL, 1996.[Ede64] Michael Edelstein, On non-expansive mappings of Banach spaces, Proc. Cambridge Philos. Soc.

60 (1964), 439�447.[ER06] Igor V. Erovenko and Andrei S. Rapinchuk, Bounded generation of S-arithmetic subgroups of

isotropic orthogonal groups over number �elds, J. Number Theory 119 (2006), no. 1, 28�48.[Eym72] Pierre Eymard, Moyennes invariantes et représentations unitaires, Springer-Verlag, Berlin, 1972,

Lecture Notes in Mathematics, Vol. 300.[Far98] Benson Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998), no. 5, 810�840.[Far08a] Benson Farb, Group actions and Helly's theorem, Preprint, 2008.[Far08b] Daniel Farley, The action of Thompson's group on a CAT(0) boundary, Groups Geom. Dyn. 2

(2008), no. 2, 185�222.[FHT08] Benson Farb, G. Christopher Hruska, and Anne Thomas, Problems on automorphism groups of

nonpositively curved polyhedral complexes and their lattices, To appear in Festschrift in honor ofRobert Zimmer's 60th Birthday, 2008.

[FL06] Thomas Foertsch and Alexander Lytchak, The de Rham decomposition theorem for metric spaces,GAFA (to appear), arXiv:math/0605419v1, 2006.

[FNS06] Koji Fujiwara, Koichi Nagano, and Takashi Shioya, Fixed point sets of parabolic isometries ofCAT(0)-spaces, Comment. Math. Helv. 81 (2006), no. 2, 305�335.

[FW06] Benson Farb and Shmuel Weinberger, Isometries, rigidity and universal covers, Annals of Math.(to appear), 2006.

[GM07] Yair Glasner and Nicolas Monod, Amenable actions, free products and a �xed point property, Bull.Lond. Math. Soc. 39 (2007), no. 1, 138�150.

[Gro87] Mikhaïl Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8,Springer, New York, 1987, pp. 75�263.

[Gro93] , Asymptotic invariants of in�nite groups, Geometric group theory, Vol. 2 (Sussex, 1991)(Cambridge), London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, 1993,pp. 1�295.

[GW71] Detlef Gromoll and Joseph A. Wolf, Some relations between the metric structure and the algebraicstructure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc.77 (1971), 545�552.

[Hag] Frédéric Haglund, Finite index subgroups of graph products, Geom. Dedicata (to appear).[Hag98] , Réseaux de Coxeter-Davis et commensurateurs, Ann. Inst. Fourier (Grenoble) 48 (1998),

no. 3, 649�666.[Hei74] Ernst Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23�34.[HLS00] Christoph Hummel, Urs Lang, and Viktor Schroeder, Convex hulls in singular spaces of negative

curvature, Ann. Global Anal. Geom. 18 (2000), no. 2, 191�204.[HM79] Roger E. Howe and Calvin C. Moore, Asymptotic properties of unitary representations, J. Funct.

Anal. 32 (1979), no. 1, 72�96.


[J�06] Tadeusz Januszkiewicz and Jacek �wi�atkowski, Simplicial nonpositive curvature, Publ. Math. Inst.Hautes Études Sci. (2006), no. 104, 1�85.

[Ka��03] Vadim A. Ka��manovich, Double ergodicity of the Poisson boundary and applications to boundedcohomology, Geom. Funct. Anal. 13 (2003), no. 4, 852�861.

[Kar53] Fridrikh I. Karpelevi£, Surfaces of transitivity of a semisimple subgroup of the group of motionsof a symmetric space, Doklady Akad. Nauk SSSR (N.S.) 93 (1953), 401�404.

[KL06] Bruce Kleiner and Bernhard Leeb, Rigidity of invariant convex sets in symmetric spaces, Invent.Math. 163 (2006), no. 3, 657�676.

[Kle99] Bruce Kleiner, The local structure of length spaces with curvature bounded above, Math. Z. 231(1999), no. 3, 409�456.

[KM99] Anders Karlsson and Gregory A. Margulis, A multiplicative ergodic theorem and nonpositivelycurved spaces, Comm. Math. Phys. 208 (1999), no. 1, 107�123.

[Lan00] Erasmus Landvogt, Some functorial properties of the Bruhat-Tits building, J. Reine Angew. Math.518 (2000), 213�241.

[Lee00] Bernhard Leeb, A characterization of irreducible symmetric spaces and Euclidean buildings ofhigher rank by their asymptotic geometry, Bonner Mathematische Schriften, 326, Universität BonnMathematisches Institut, Bonn, 2000.

[LMR00] Alexander Lubotzky, Shahar Mozes, and Madabusi Santanam Raghunathan, The word and Rie-mannian metrics on lattices of semisimple groups, Inst. Hautes Études Sci. Publ. Math. (2000),no. 91, 5�53 (2001).

[LY72] H. Blaine Lawson, Jr. and Shing-Tung Yau, Compact manifolds of nonpositive curvature, J. Dif-ferential Geometry 7 (1972), 211�228.

[Lyt05] Alexander Lytchak, Rigidity of spherical buildings and joins, Geom. Funct. Anal. 15 (2005), no. 3,720�752.

[Mal40] Anatoly I. Mal'cev, On isomorphic matrix representations of in�nite groups, Rec. Math. [Mat.Sbornik] N.S. 8 (50) (1940), 405�422.

[Mar91] Gregory A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematikund ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991.

[MMS04] Igor Mineyev, Nicolas Monod, and Yehuda Shalom, Ideal bicombings for hyperbolic groups andapplications, Topology 43 (2004), no. 6, 1319�1344.

[Mon01] Nicolas Monod, Continuous bounded cohomology of locally compact groups, Lecture Notes in Math-ematics, vol. 1758, Springer-Verlag, Berlin, 2001.

[Mon05] , Arithmeticity vs. nonlinearity for irreducible lattices, Geom. Dedicata 112 (2005), 225�237.

[Mon06] , Superrigidity for irreducible lattices and geometric splitting, J. Amer. Math. Soc. 19(2006), no. 4, 781�814.

[Mos55] George D. Mostow, Some new decomposition theorems for semi-simple groups, Mem. Amer. Math.Soc. 1955 (1955), no. 14, 31�54.

[Mou88] Gabor Moussong, Hyperbolic coxeter groups, Ph.D. thesis, Ohio State University, 1988.[MP03] Nicolas Monod and Sorin Popa, On co-amenability for groups and von Neumann algebras, C. R.

Math. Acad. Sci. Soc. R. Can. 25 (2003), no. 3, 82�87.[MZ55] Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers,

New York-London, 1955.[Osi06] Denis Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic

problems, Mem. Amer. Math. Soc. 179 (2006), no. 843, vi+100.[Pin98] Richard Pink, Compact subgroups of linear algebraic groups, J. Algebra 206 (1998), no. 2, 438�504.[Pla65] Vladimir P. Platonov, Lokal projective nilpotent radicals in topological groups., Dokl. Akad. Nauk

BSSR 9 (1965), 573�577.[Pra77] Gopal Prasad, Strong approximation for semi-simple groups over function �elds, Ann. of Math.

(2) 105 (1977), no. 3, 553�572.[Pra82] , Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bull.

Soc. Math. France 110 (1982), no. 2, 197�202.[Rag72] Madabusi Santanam Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York,

1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68.[Rag04] , The congruence subgroup problem, Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 4,

299�308.


[Sel60] Atle Selberg, On discontinuous groups in higher-dimensional symmetric spaces., Contrib. FunctionTheory, Int. Colloqu. Bombay, Jan. 1960, 147-164 (1960)., 1960.

[Ser70] Jean-Pierre Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. (2) 92 (1970),489�527.

[Ser77] , Arbres, amalgames, SL2, Société Mathématique de France, Paris, 1977, Avec un sommaireanglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46.

[Sha00] Yehuda Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000),no. 1, 1�54.

[Sim96] Michel Simonnet,Measures and probabilities, Universitext, Springer-Verlag, New York, 1996, Witha foreword by Charles-Michel Marle.

[Swe99] Eric L. Swenson, A cut point theorem for CAT(0) groups, J. Di�erential Geom. 53 (1999), no. 2,327�358.

[Tak02] Masamichi Takesaki, Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences,vol. 124, Springer-Verlag, Berlin, 2002, Reprint of the �rst (1979) edition, Operator Algebras andNon-commutative Geometry, 5.

[Tav90] Oleg I. Tavgen′, Bounded generability of Chevalley groups over rings of S-integer algebraic num-bers, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 97�122, 221�222.

[Tho06] Anne Thomas, Lattices acting on right-angled buildings, Algebr. Geom. Topol. 6 (2006), 1215�1238(electronic).

[Thu97] William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton MathematicalSeries, vol. 35, Princeton University Press, Princeton, NJ, 1997, Edited by Silvio Levy.

[Tit64] Jacques Tits, Algebraic and abstract simple groups, Ann. of Math. (2) 80 (1964), 313�329.[Tit72] , Free subgroups in linear groups, J. Algebra 20 (1972), 250�270.[Tit87] , Uniqueness and presentation of Kac-Moody groups over �elds, J. Algebra 105 (1987),

no. 2, 542�573.[U²a63] V. I. U²akov, Topological FC-groups, Sibirsk. Mat. �. 4 (1963), 1162�1174.[Vin94] Èrnest B. Vinberg (ed.), Lie groups and Lie algebras, III, Encyclopaedia of Mathematical Sciences,

vol. 41, Springer-Verlag, Berlin, 1994, Structure of Lie groups and Lie algebras, A translation ofCurrent problems in mathematics. Fundamental directions. Vol. 41 (Russian), Akad. Nauk SSSR,Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, Translation by V. Minachin.

[Weh73] Bertram A. F. Wehrfritz, In�nite linear groups. An account of the group-theoretic properties ofin�nite groups of matrices, Springer-Verlag, New York, 1973, Ergebnisse der Matematik und ihrerGrenzgebiete, Band 76.

[Wil71] John S. Wilson, Groups with every proper quotient �nite., Proc. Camb. Philos. Soc. 69 (1971),373�391.

[Wis04] Daniel T. Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004), no. 1, 150�214.

[WM07] Dave Witte Morris, Bounded generation of SL(n, A) (after D. Carter, G. Keller, and E. Paige).,New York J. Math. 13 (2007), 383�421.

[Wor07] Kevin Wortman, Quasi-isometric rigidity of higher rank S-arithmetic lattices., Geom. Topol. 11(2007), 995�1048.

[Yau82] Shing Tung Yau, Problem section, Seminar on Di�erential Geometry, Ann. of Math. Stud., vol.102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669�706.

[Zim78] Robert J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries ofrandom walks, J. Functional Analysis 27 (1978), no. 3, 350�372.

[Zim84] , Ergodic theory and semisimple groups, Birkhäuser Verlag, Basel, 1984.[Zim87] , Amenable actions and dense subgroups of Lie groups, J. Funct. Anal. 72 (1987), no. 1,

58�64. MR MR883503 (89a:22012)

Index

admissible group, 60algebraic action, 34amenable radical, see radicalangleAlexandrov, 13asymptotic formula, 14comparison, 13rigidity, 26, 40Tits, 13

antipode, 41arithmetic group, 60ascending, 6, 20

Bass�Serre tree, 39Borel density, 10, 43, 50boundary-minimal, see minimalBusemanncharacter, 18function, 13

canonicaldiscrete kernel, 55quotient, 55

CAT(0) group, 3CAT(0) lattice, 2arithmetic, 3uniform, 3

CAT(0) spacegeodesically complete, 24

circumcentre, 15canonical, 15intrinsic, 15

circumradius, 15intrinsic, 15

co-amenable, 42cône, 22parabolic, 35topology, 14

discrete residual, see residual

Euclideanfactor, 14pseudo-factor, 33

extensible geodesic, 24

factor(pseudo-)Euclidean, see Euclideanvon Neumann algebra, 45

�nite residual, see residual

geodesically complete, see CAT(0) space

hull, 54of a lattice, 54

irreducible

group, 4, 52lattice, 52

isometryballistic, 18neutral parabolic, 11, 72parabolic, 11

LF-radical, see radicallimit set, 45locally �nite group, 29

minimalCAT(0) space, 5action, 5boundary-, 5

monolith, 12, 28monolithic, 12, 28

neutral, see isometry

opposite, 33orderof an ascending subgroup, 20

parabolic, see isometryparabolic cône, see côneproper space, 13pseudo-Euclidean factor, see Euclidean

quasi-centre, 7, 29quasi-simple, 28

radicalamenable, 4LF-, 29superrigidity, 65

reduced action, 17residualdiscrete, 55�nite, 11, 69

Schreier graph, 28smooth, 8, 24, 77socle, 7, 32space of directions, 38

Thompson's group, 42Titsangle, see angleboundary, 14distance, 14, 33

translation length, 11

84


IHÉS, Route de Chartres 35, 91440 Bures-sur-Yvette, FranceE-mail address: [email protected]

EPFL, 1015 Lausanne, SwitzerlandE-mail address: [email protected]

ISOMETRY GROUPS AND LATTICES OF NON-POSITIVELY CURVED...

Documents

Transcript of ISOMETRY GROUPS AND LATTICES OF NON-POSITIVELY CURVED...