RIGIDITY AND ARITHMETICITY

Marc BURGER

1. Introduction. Statements of the results

Lattices. Mostow’s rigidity theorem. Borel-Harish Chandra’s theorem. Margulis’ arithmeticitytheorem. Margulis’ arithmeticity criterion. Margulis’ superrigidity theorem.

2. Invariant measures on homogeneous spaces

Existence and uniqueness of Haar measures. Integration formulas. Group actions on functionsand measures on a locally compact space. Existence and uniqueness of semi-invariant measureson homogeneous spaces. Applications to lattices. Morphisms of homogeneous spaces.

3. Examples of arithmetic lattices

3.A The inclusion SLn(Z) < SLn(R).

Minkowski’s theorem. The group SLn(Z) is a lattice. Mahler’s compactness criterion.

3.B Quaternion algebras and arithmetic lattices of SL2(R)

Quaternion algebras. Cocompactness and arithmeticity of maximal orders.

3.C. Restriction of scalars and arithmetic lattices

Almost k-simple groups. Lie and algebraic groups. Restriction of scalars.

4. Amenable groups

Amenable groups via Frechet spaces. Abelian implies amenable. Extensions and amenability.Solvable implies amenable. Compact implies amenable. Minimal parabolic subgroups areamenable. Existence of invariant probability measures on a compact space.

5. Furstenberg maps

Existence of Γ-equivariant maps G/P →M1(PVk). Connection with Teichmuller theory.

6. The Zariski Support map

6.A The Support map is Borel and PGL(Vk)-equivariant.

6.B The Zariski Closure map is Borel and PGL(Vk)-equivariant.

6.C The Zariski Support map M1(PVk)→ Vark(PV ) is Borel and PGL(Vk)-equivariant.

7. Ergodic actions and Borel structures

7.A Ergodic actions

Equivalence of positive Radon measures. Invariant classes. Ergodic actions. Actions of somesemisimple elements in simple groups. Unitary representations. Ergodicity of a lattice actionon G/P ×G/P .

1

7.B Borel structures

Borel spaces and σ-separated Borel spaces. Invariant measurable maps with ergodic sourceand σ-separated targets. Existence of a conull orbit. Locally closed orbits and σ-separatedBorel structures. Topological condition for local closedness of orbits.

8. The boundary map. The first case of dichotomy

8.A The boundary map Φ = SuppZ ◦ ϕ.

8.B Dynamics of PGL(Vk) on PVk. Furstenberg’s lemma.

8.C Stabilizers of measures. If Φ is essentially constant, then π(Γ) relatively compact.

8.D The case of SL2(R).

9. The boundary map. The second case of dichotomy

If Φ is not essentially constant, then there is a Γ-equivariant map G/P → Hk/Lk.

10. Commensurator Superrigidity

Construction of a Λ-equivariant map for Γ < Λ < CommG(Γ).

11. Higher Rank Superrigidity

Essentially rational maps. Continuous extensions of homomorphisms of abstract groups.

12. Arithmeticity

Arithmetic presentation in the higher rank case.

2

1. INTRODUCTION. STATEMENT OF THE MAIN RESULTS

The aim of these lectures is to expose the proof of a fundamental result due to Margulis,concerning the structure of lattices in semisimple Lie groups: the arithmeticity theorem. Thischapter is devoted to the descrption of related results and of the main objects involved. Manyof the notions used here will be introduced in more detail in subsequent chapters.

DEFINITION.— Let G be a locally compact group. A subgroup Γ of G is called a lattice if(i) it is discrete; equivalently, any compact subset of G meets Γ in finitely many points;(ii) the homogeneous space G/Γ admits a finite nonzero G-invariant measure.

REMARKS.— 1) In these lectures, a measure on a locally compact space will always refer to aregular Borel measure which is finite on compact sets.2) As a consequence of a discussion of the existence of invariant measures on homogeneousspaces, we will see that if Γ is a lattice in G, the G-invariant measure on G/Γ is unique up toa scaling factor.

DEFINITION.— A closed subgroup H < G is called cocompact if G/H is compact.

EXAMPLES.— 1) G = Rn, Γ = Zn. More generally, a1,... an being a basis of Rn, the Z-module Λ := Za1 + ... + Zan is a lattice in Rn. Observe that Rn/Λ is compact. It is a factthat all lattices of Rn are obtained this way. In particular, let Λ and Λ′ be lattices in Rn, andA : Λ→ Λ′ be any group isomorphism. Then A extends to a continuous automorphism of Rn

(an invertible linear map) Aext which obviously satisfies Aext(Λ) = Λ′. Therefore, Zn is up tocontinuous automorphisms of Rn, the unique lattice in Rn.

2) G = {

1 x z0 1 y0 0 1

: x, y, z ∈R} is a 2-step nilpotent Lie group – called the Heisenberg

group – and the subgroup Γ = {

1 x z0 1 y0 0 1

: x, y, z∈Z} is a lattice in G. Observe that G/Γ

is compact.3) G = SLn(R), Γ = SLn(Z). Here is a description of the homogeneous space SLn(R)/SLn(Z).Let R denote the set of all lattices in Rn. The group GLn(R) acts transitively on R – seeexample 1) – and the stabilizer of the point Zn∈R is GLn(Z). Hence we obtain a bijection

GLn(R)/GLn(Z)∼→ R

g 7→ gZn.

From this we deduce that if R1 := {Λ∈R : Vol(Rn/Λ) = 1}, then SLn(R) acts transitivelyon R1 and we obtain a bijection

SLn(R)/SLn(Z)∼→ R1

g 7→ gZn.

We will see in section 3.A that SLn(Z) is a lattice in SLn(R).

Given a connected simple group G, the main problem is to get a classification of all latticesin G. Concerning the degree of generality adopted here, we will always think of G as being aclassical group; in particular, it will always be a real matrix group. Concerning the classificationproblem, a basic question is: does the group structure of Γ determine its position in G up to

3

automorphisms of G ? In other words, are two isomorphic lattices in G conjugate by anautomorphism of the Lie group G ? The answer is given by Mostow’s rigidity theorem.

THEOREM (Mostow).— Let G, G′ be connected simple Lie groups with trivial center andΓ < G and Γ′ < G′ be lattices. Assume rk(G) ≥ 2. Then any group isomorphism

θ : Γ∼−→ Γ′

extends to an isomorphism of Lie groups

θext : G∼−→ G′.

REMARK. —This theorem does not hold for PSL2(R). The notion of rank is essential there.

EXAMPLES.— 1) The group SLn(R) is of rank n−1, it is the dimension of the group of diagonalmatrices with positive entries and determinant 1, which is a maximal R-split torus.2) G = SO(F ), the special orthogonal group of a quadratic form F : Rn → R of index p. In asuitable basis e1, ... en, F has matrix J

F0

J

,

where J := [δi+j,p+1] is a p × p matrix and F0 : Rn−2p → R is positive definite. As maximalR-split torus, one may choose

S = {

λ1

...λp

In−2p

λ−1p

...λ−1

1

: λi 6= 0}

whose centralizer is SO(F0). Here rk(SO(F )

)= p.

Although this theorem shows that the lattice Γ determines the ambient Lie group, it doesnot provide a method to construct lattices. The fundamental result of Margulis is that inLie groups G of the type occuring in Mostow’s theorem, all lattices are obtained by an〈〈arithmetic 〉〉 construction. The Borel-Harish Chandra’s theorem justifies the main step ofthis construction.

THEOREM (Borel-Harish Chandra).— Let G < GLn be an algebraic group defined over Q,which is semisimple as a Lie group. Then G(Z) := G(Q) ∩GLn(Z) is a lattice in GR.

EXAMPLE.— Set F = x21 + x2

2 − px23, for p a prime. Define also

GF = SO(F ) = {g∈GL3 : tg

11−p

g =

11−p

, detg = 1}.

This is a simple algebraic group defined over Q, and GF (R) is conjugate inside GL3(R) toSO2,1(R). Fixing such an isomorphism

4

j : GF (R)∼−→ SO2,1(R),

we obtain a lattice ΓF := j(GF (Z)

)∈ SO2,1(R). Varying p, we obtain many non conjugate

lattices in SO2,1(R).

This example suggests that arithmetic lattices in a simple Lie group G should be latticesobtained via algebraic groups defined over Q, whose real points are isomorphic to G as a Liegroup. There is however a slight generalization of this construction.

LEMMA.— Let G, H be locally compact groups, Γ a lattice in H and p : H → G a surjectivecontinuous homomorphism with compact kernel. Then p(Γ) is a lattice in G.

Proof.— The map p is proper, therefore p(Γ) is discrete in G. The map p induces a continuousmap

ϕ : H/Γ→ G/p(Γ)

which equivariant, i.e., ϕ(hx) = p(h)ϕ(x), for all h ∈ H and x ∈ H/Γ. If α denotes anH-invariant measure on H/Γ, the direct image ϕ∗α is then a G-invariant finite measure onG/p(Γ). �

REMARK.— In the previous lemma, the compactness condition on the kernel is essential.

Indeed, take H = R2, Γ = Z

(11

)⊕Z

(0α

)with α∈R \Q. Set also G = R and p : R2 → R

second projection. Then p(Γ) is dense in G.

DEFINITION.— Let G be a group and Γ and Γ′ be subgroups of G. Then Γ and Γ′ are calledcommensurable if Γ ∩ Γ′ is of finite index in Γ and Γ′.

REMARK.— Let Γ be a lattice in G locally compact, and Γ′ be commensurable with Γ. ThenΓ′ is a lattice in G. This follows from the following diagram

G/(Γ ∩ Γ′) → G/Γ↓

G/Γ′

And now (at last) the notion of an arithmetic lattice.

DEFINITION.— Let G be a simple Lie group. A lattice Γ∈G is called arithmetic if there existan algebraic semisimple group H defined over Q and a continuous surjective homomorphism

p : H(R)◦ → G

with compact kernel such that p(H(Z) ∩H(R)◦

)and Γ are commensurable. The pair (p,H)

will be referred to as an arithmetic presentation of Γ.

THEOREM (Margulis).— Let G be a connected simple Lie group with trivial center and rank≥ 2. Every lattice in G is arithmetic.

The proof of this theorem is based on the classification of irreducible linear representations ofΓ, of which Mostow’s rigidity theorem is a special case. While all lattices in higher rank arearithmetic, it has been shown for a long time that this is not true in PSL2(R); Gromov andPiatetski-Shapiro have shown that there exist non-arithmetic lattices in all groups SOn,1(R),n ≥ 2. So the question is, given a lattice, find a necessary and sufficient condition for Γ to bearithmetic. Margulis found a criterion which is based on the commensurator of Γ∈G.

5

DEFINITION.— The set CommG(Γ) := {g∈G : Γ and gΓg−1 are commensurable} is a subgroupcontaining Γ and called the commensurator subgroup of Γ∈G.

EXAMPLE.— Set G = SL2(R) and Γ = SL2(Z). Then CommG(Γ) > SL2(Q). Indeed, letg∈SL2(Q) and m be the smallest common multiple of all denominators of the entries of g andg−1. Let

Γ′ := {γ∈SL2(Z) : γ ≡ idmodm2}.

Clearly Γ′ is a subgroup of finite index in SL2(Z). Furthermore, if γ = id + m2B is in Γ′,where B has integral entries, then

gγg−1 = id + (mg)B(mg)−1 ∈ SL2(Z).

Therefore gΓ′g−1 < SL2(Z) and gSL2(Z)g−1 ∩ SL2(Z) is of finite index in SL2(Z) since itcontains gΓ′g−1.

The criterion of Margulis is the following.

THEOREM (Margulis).— Let G be a connected simple Lie group with trivial center. A latticeΓ∈G is arithmetic if and only if CommG(Γ)/Γ is infinite.

REMARKS.— 1) This applies in arbitrary rank.2) If Γ is not arithmetic, then CommG(Γ)/Γ is finite and therefore CommG(Γ) is a lattice inG. In fact, it is the unique maximal lattice of G containing Γ.

The arithmeticity results will be a direct consequence of Margulis’ superrigidity theorem. Mostof these lectures will be devoted to the proof of this theorem.

THEOREM (Margulis).— Let G be a connected almost R-simple algebraic group,Γ < G := G◦R be a lattice, and Λ be a subgroup such that

Γ < Λ < CommGΓ.

Let k be a local field of characteristic zero, H an adjoint connected almost k-simple group,and

π : Λ→ Hk

a homomorphism such that π(Γ) is Zariski dense in H and unbounded in Hk. Assume eitherthat Λ is dense in G or that G is of rank ≥ 2. Then k is R or C and π extends to ak-homomorphism of algebraic groups:

πext : G→ H.

2. INVARIANT MEASURES ON HOMOGENEOUS SPACES

Let G be a locally compact group. One has the following basic result due to Haar.

THEOREM 2.1.— On G there exists, up to a positive scaling factor, a unique left invariantpositive Radon measure.

REMARKS.— 1) On a Lie group G, such a Haar measure is easily constructed via a nonzeroleft invariant differential n-form (for n = dimG).

6

2) We will frequently use Riesz’s representation theorem which identifies continuous linearfunctionals on K(X), the space of continuous functions with compact support, with Radonmeasures on X (X locally compact).

Let dg be a Haar measure on G. The following two facts are immediate consequences of theuniqueness statement in the above theorem.

1) There is a continuous homomorphism

∆G : G→ R×+

such that ∫G

f(gh−1)dg = ∆G(h)

∫G

f(g)dg

h∈G, f ∈K(G). The homomorphism ∆G doesn’t depend on the choice of dg.

The group G is called unimodular if ∆G is identically equal to 1, that is any left invariantmeasure is also right invariant.

2) One has: ∫G

f(g−1)∆G(g−1)dg =

∫G

f(g)dg,

for all f ∈K(G).

EXAMPLES. —1) Lebesgue measure on Rn.2) A connected semisimple Lie group has no nontrivial homomorphism to R×+ and is thereforeunimodular. The same argument shows that so is a compact group.3) Let N be the group of n×n upper triangular matrices with 1’s on the diagonal. N = {[xij ] :xii = 1 and xij = 0} for i > j. Then

∏i<j dxij is a Haar measure on N . Indeed, if A and X

are in N , then (AX)ij = Xij + terms not involving Xij . The same argument shows that themeasure is right invariant, hence N is unimodular.4) Let B+ be the group of n×n upper triangular matrices with positive entries on the diagonaland determinant 1. Then B+ = A+ n N , where A+ is the group of diagonal matrices withpositive entries and determinant =1. Let da+ be a Haar measure on A+, dn be a Haar measureon N . Then db+ := da+dn is a Haar measure on B+. Indeed∫

A+

da+

∫N

dnf(a′n′an) =

∫A+

da+

∫N

dnf(a′an) =

∫A+

da+

∫N

dnf(an),

use a′n′an = a′a(a−1n′a)n. One may check that ∆B+ =∏i<j

aiaj

.

5) The counting measure on a discrete group is a Haar measure.

NOTATIONS. —Let G be a group and X a set on which G acts on the left. Let Y be a set andF the set of maps X → Y . Then G acts on the left on F via

g∗f(x) = f(g−1x).

In particular, if X is locally compact and G acts by homeomorphisms on X, then G acts onK(X), and on the space M(X) of Radon measures:

7

g∗f(x) = f(g−1x), f ∈K(X),g∗µ(f) = µ(g−1

∗ f), µ∈M(X).

Let H be a closed subgroup of G.

DEFINITION.— A positive Radon measure µ on G/H is called semi-invariant if g∗µ and µ areproportional for all g∈G. The proportionality factor defines then a continuous homomorphismχ : G→ R∗× called the modulus of µ.

THEOREM 2.2.— Let χ : G → R∗× be a continuous homomorphism. There exists a semi-

invariant measure of modulus χ on G/H if and only if χ |H=∆G |H∆H

. This measure is unique

up to a positive scaling factor and after normalizing dg one has∫G/H

dµ(x)

∫H

f(xh)dh =

∫G

f(g)χ(g)−1dg, f ∈K(G).

Reference. See for example [R], preliminaries. �

COROLLARY 2.3.— If G and H are both unimodular, then there exists ( up to scaling ) a uniqueG-invariant measure on G/H. This happens in particular when G is connected simple and His discrete. �

As an application of corollary 2.3, we compute the Haar measure on SLn(R). Let B+ be asin example 4) and K = SOn(R). One has the Iwasawa decomposition

SLn(R) = B+ ·K

(this follows from Gram-Schmidt orthonormalization). We have therefore a B+-equivariantidentification SLn(R)/K → B+. Since both SLn(R) and K are unimodular, there is a SLn(R)-invariant measure dµ on SLn(R)/K. Via the above identification, it must correspond to a leftHaar measure db+ on B+. It follows then from theorem 2.2 and corollary 2.3 that∫

B+

db+∫K

dkf(b+k)

is a Haar measure on SLn(R).

Finally we have another immediate consequence of theorem 2.2.

COROLLARY 2.4.— Let H1 < H2 be closed subgroups of G and assume that H1, H2 and G areunimodular. There is a choice of invariant measures on G/H1, G/H2 and H2/H1 such that∫

G/H1

f(y)dy =

∫G/H2

dx

∫H2/H1

f(xz)dz.

�

3. EXAMPLES OF ARITHMETIC LATTICES

3.A SLn(Z) < SLn(R)

First we need the following basic result due to Minkowski.

PROPOSITION 3.1 (Minkowski’s lemma).— Let Λ ⊂ Rn be a lattice and V ⊂ Rn be a closedbounded balanced convex set such that: Vol(V ) ≥ 2n ·Vol(Rn/Λ). Then: V ∩ (Λ \ {0}) 6= ∅.

8

Proof.— Let π : Rn → R/Λ be the canonical projection.

Claim: π is not injective on 12V .

Assume the contrary. Then:

(?) Vol(π( 1

2V ))

= Vol(

12V)≥ Vol(Rn/Λ).

Observe that π( 12V ) ( Rn/Λ. Otherwise, one would have

(??) Rn =⊔λ∈Λ

(λ+1

2V ).

This union is disjoint because π is injective on 12V . Now a pathwise connected topological

space cannot be a countable union of proper closed disjoint subsets. So (??) is impossible,hence π( 1

2V ) ( Rn/Λ. But then Vol(π( 1

2V ))< Vol(Rn/Λ), which contradicts (?). Hence, π

is not injective on 12V .

Therefore there exist x ∈ 12V and λ ∈ Λ, λ 6= 0 with x + λ ∈ 1

2V . Since 12V is convex and

balanced, one has − 12x+ 1

2 (x+ λ)∈ 12V , and therefore λ∈V . �

THEOREM 3.2 —SLn(Z) is a lattice in SLn(R).

Proof.— By induction on n. For n = 1, this is clear. Now assume n ≥ 2 and that SLn−1(Z)is a lattice in SLn−1(R). Consider the transitive action of SLn(R) on Rn \ {0}. Then

N := Stab

10· · ·0

= {

1 ∗0· · ·0

∗ · · · ∗· · · ∗ · · ·∗ · · · ∗

∈ SLn(R)}

and NZ = SLn(Z) ∩N , so that N = SLn−1(R) n Rn−1 and NZ = SLn−1(Z) n Zn−1.

By induction hypothesis, SLn−1(Z) is a lattice in SLn−1(R). Form this follows that NZ is alattice in N . Consider the diagram

SLn(R)/NZ → SLn(R)/N ∼= Rn \ {0}↓

SLn−1(R)/SLn−1(Z)

Let f be an integrable function on Rn \ {0}. Since Vol(N/NZ) <∞, the function

SLn(R)/NZ → Rg 7→ f(ge)

is integrable and – see corollary 2.4:∫SLn(R)/NZ

f(ge)dg = Vol(N/NZ)

∫Rn\{0}

f(x)dx.

For the same reason, we have∫SLn(R)/SLn(Z)

∑γ∈SLn(Z)/NZ

f(gγe) =

∫SLn(R)/NZ

f(ge)dg.

9

Hence, setting F (g) :=∑

γ∈SLn(Z)/NZ

f(gγe), we have

(?)

∫SLn(R)/SLn(Z)

F (g)dg = Vol(N/NZ)

∫Rn

f(x)dx.

Observe that the orbit under SLn(Z) of the first canonical vector of Zn is the set of vectorswhose coordinates have gcd=1. Recall that SLn(R)/SLn(Z) may be identified with R1, theset of lattices Λ < Rn with Vol(Rn/Λ) = 1. Hence, we consider F as a function R1 → R.Then it follows that

F (Λ) =∑

λ∈Λ\{0},primitive

f(λ)

Now, if f is the characteristic function of [−1; 1]n, it follows from proposition 3.1 that F (Λ) ≥ 1for any Λ∈R1. Hence, using (?), we obtain

Vol(SLn(R)/SLn(Z)

)≤∫

SLn(R)/SLn(Z)

F (g)dg = 2n ·Vol(N/NZ) <∞

�

Now we turn to the question of the characterisation of relatively compact subsets in SLn(R)/SLn(Z).Indeed, as we will see below, although SLn(R)/SLn(Z) has finite invariant volume, this spaceis not compact. We will take

R1 := {Λ ⊂ Rn : Λ lattice Vol(Rn/Λ) = 1}

(see proof above) as a model of SLn(R)/SLn(Z).

One has the following intrinsic description of the topology on R1. Let B ⊂ Rn be any openball centered at the origine and define for Λ, Λ′∈R1

dB(Λ,Λ′) := maxλ∈Λ∩Bλ′∈Λ′∩B

{d(λ,Λ′ ∩B), d(λ′,Λ ∩B)}

where d is the Euclidean distance of Rn. Then a sequence (Λn)n∈Z converges to Λ′ if and onlyif

limn→∞

dB(Λn,Λ′) = 0

for every ball B ⊂ Rn.

THEOREM 3.3 (Mahler’s compactness criterion) —A subset M ⊂ R1 is relatively compact ifand only if there exists U ⊂ Rn neighborhood of {0} such that

Λ ∩ U = {0} ∀Λ∈M .

Proof.— (⇒) Assume that M is compact. Let B = B(0, R) be the closed ball of radius R > 0and center 0. The map

R1 → NΛ 7→ #(Λ ∩B)

is upper semicontinuous. In fact, for every Λ∈R1, there is a neighborhood VΛ ⊂ R1 such that

#(Λ′ ∩B) ≤ #(Λ ∩B)

10

for all Λ′ ∈ VΛ (geometrically clear). From the assumption that M is compact follows thatfinitely many such neighborhoods VΛ cover M . Hence Λ 7→ #(Λ∩B) is bounded on M . Fromthis follows easily that there exists Br = B(0, r) such that

Λ ∩Br = {0} ∀Λ∈M .

(⇐) Assume that there exists r > 0 such that

Λ ∩B(0, r) = {0} ∀Λ∈M .

To every lattice Λ ⊂ Rn, we associate a closed convex subset CΛ ⊂ Rn defined by

CΛ := {x∈Rn : d(x, 0) ≤ d(x, λ) ∀λ∈Λ}.

Then Cλ is the intersection of the family of half-spaces

Hλ := {x∈Rn : d(x, 0) ≤ d(x, λ)}, λ∈Λ.

Claim 1.— Cλ is a fundamental domain for the action of Λ on Rn.

(1) Any point x ∈Rn is equivalent mod Λ to a point in Cλ; equivalently, π : Rn → Rn/Λrestricted to CΛ is surjective.

Indeed, since Λ is discrete, the set {d(x, µ) : µ∈Λ} has a minimum, say d(x, λ) for some λ∈Λ.The distance d being translation invariant, we obtain x− λ∈Cλ.

(2) Two Λ-equivalent points in Cλ lie on the boundary ∂Cλ; equivalently, π : Rn → Rn/Λrestricted to the interior of CΛ is injective.

Indeed, assume x∈Cλ and x+ λ ∈Cλ for some λ 6= 0. Then:

d(x, 0) ≤ d(x,−λ) = d(x+ λ, 0) ≤ d(x+ λ, λ) = d(x, 0),

hence d(x, 0) = d(x,−λ), or x∈Hλ. Since ∂Cλ =⋃λ∈Λ

(Cλ ∩ ∂Hλ), this proves (2).

Claim 2.— There is a constant c(r) > 0, only depending on r, such that for all Λ∈M we have

Cλ ⊂ B(0, c(r)

).

Since Cλ is a fundamental domain for Λ ⊂ Rn, we have Vol(Cλ) = Vol(Rn/Λ) = 1. Observealso that if Λ ∩ B(0, r) = {0}, then B(0, r/2) ⊂ CΛ. Therefore, for every x∈CΛ, the convexhull of x and B(0, r/2) is of volume ≤ 1. Hence ‖ x ‖≤ C(r) for some constant c(r). Thisproves claim 2.

In particular, Cλ =⋃λ∈SΛ

Hλ where SΛ is the finite set SΛ := Λ ∩ B(0, 2c(r)

). Since for all

Λ∈M the points of Λ are distance r/2-apart, the cardinality of SΛ is bounded by the constant

κ(r) :=Vol(B(0, 2c(r) + r/2)

)Vol(B(0, r/2)

) ,

which depends only on r > 0.

Claim 3.— SΛ generates Λ.

Let Λ′ be the subgroup generated by SΛ, and CΛ′ the corresponding fundamental domain. Forx∈CΛ′ we have

11

d(x, 0) = minλ′∈Λ′

d(x, λ′) ≤ minλ∈Λ

d(x, λ)

and hence x∈CΛ. Therefore CΛ′ ⊂ CΛ and

Vol(Rn/Λ′) = Vol(CΛ′) = Vol(Rn/Λ).

On the other hand, Λ′ is a subgroup of Λ and therefore

Vol(Rn/Λ′) = #(Λ/Λ′) ·Vol(Rn/Λ).

From these, we conclude that #(Λ/Λ′) = 1, and hence Λ = Λ′. This prove claim 3.

Now we show that M is relatively compact by proving that any sequence (Λn)n∈N of M has

a convergent subsequence in R1. Since #SΛn is bounded by κ(r), SΛn ⊂ B(0, 2c(r)

), we may

assume up to passing to a subsequence, that

limn→∞

SΛn = S

where S ⊂ B(0, 2c(r)

)is a finite set. This convergence is to be understood in the following

sense.

(1) #SΛn = #S for all integer n.

(2) Any ε-neighborhood of S contains SΛn for n big enough.

Now let Λ be the Z-module generated by S. We claim that Λ is a lattice in Rn and

limn→∞

Λn = Λ.

(a) For n big enough, we have a bijection

bn : S∼−→ Sn

such that for every λ∈S, d(λ, bn.λ)→ 0 as n→∞. Now for λ =∑µ∈S

mµµ∈Λ, set

λµ :=∑µ∈Smµbn(µ) ∈ Λn.

Then limn→∞

λn = λ. In particular, if ‖ λ ‖< r we have ‖ λn ‖< r for n big and hence λn = 0,

since Λn ∩B(0, r) = {0}. So λ = 0. This shows that Λ is discrete.

(b) The rank of Λ is n. Let V be the vector space generated by S. Let x∈Rn be any vectororthogonal to V . Then we have

d(x, 0) < d(x, λ), ∀λ∈S.

Hence for n big enough we have d(x, 0) < d(x, µ) for all µ ∈ SΛn . In particular x ∈ CΛn ⊂B(0, 2c(r)

). So any vector x orthogonal to V is of length ≤ 2c(r). This implies clearly that

V = Rn.

The proof that limn→∞

Λn = Λ is left to the reader as an exercise. �

12

3.B Quaternion algebras and arithmetic lattices in SL2(R)

In this section we will construct cocompact arithmetic lattices in SL2(R) using quaternionsalgebras over Q. Let K be a field with Q ⊂ K ⊂ C and a, b∈Z nonzero integers. Let

Ha,b(K) := K1⊕Ki⊕Kj ⊕Kk.

be the 4-dimensional K-vector space with basis {1; i; j; k}. There is an associative K-algebrastructure on Ha,b(K) whose product is defined on the basis by:

k = ij = −ji, i2 = a and j2 = b.

Ha,b(K) is called a quaternion algebra over K.

On Ha,b(K) we have an involution x = x1 + x2i + x3j + x4k 7→ x := x1 − x2i − x3j + x4ksatisfying x.y = y.x, and a norm

N : Ha,b(K)→ K

sending x to xx = x21 − x2

2a − x23b + x2

4ab and which is multiplicative: N(xy) = N(x)N(y).Let Ha,b(K)× denote the group of invertible elements of Ha,b(K). Observe that x∈Ha,b(K)×

if and only if N(x) 6= 0, in which case x−1 = x/N(x). In particular, Ha,b(K) is a divisionalgebra if and only if N−1(0) = {0}.

EXAMPLES. —1) If K = R and a = b = −1, one gets the Hamilton quaternion algebra.2) K = Q, a = b = 1. Ha,b(Q) is not a division algebra since N(1 + i+ j + k) = 0.3) K = Q, b is a prime number, and a is not a square modulo b. Then Ha,b(Q) is a divisionalgebra. Indeed, assume that N(x) = 0 for some nonzero x ∈ Ha,b(Q). We may assumealso that xi are integers, with gcd=1. Reducing x2

1 − x22a − x2

3b + x24ab modulo b, one gets

x21 = x2

2a ∈ Z/bZ. Hence x1 = x2 = 0 ∈ Z/bZ since b is not a square modulo b. Thereforex1 = y1b and x2 = y2b for yi∈Z and

y21b

2 − y22ab

2 − x23b+ x2

4ab = 0

hence

y21b− y2

2ab− x23 + x2

4a = 0.

Reducing modulo b, we get x24a = x2

3∈Z/bZ, and so x3 = x4 = 0∈Z/bZ, contradicting gcd=1.

NOTATIONS. —If A is any subring of C with unity, Ha,b(A) will denote the ring with unityA1⊕Ai⊕Aj ⊕Ak. It is invariant under x 7→ x, so

H1a,b(A) := {x∈Ha,b(A) : N(x) = 1}.

is a group.

For the rest of this paragraph, we assume that a and b are positive integers. One verifies thatthe map

h : Ha,b(R)→M2,2(R)

x 7→(

x1 + x2√a x3

√a+ x4

√ab

x3√a− x4

√ab x1 − x2

√a

)is an isomorphism of R-algebras with N(x) = deth(x).

13

Therefore h induces the following Lie groups isomorphisms

h : Ha,b(R)× → GL2(R);

h : H1a,b(R)→ SL2(R).

Now H1a,b(Z), being the intersection of the discrete set Ha,b(Z) and the closed subgroup

H1a,b(R), is a discrete subgroup of H1

a,b(R). Set Γa,b := h(H1a,b(Z)

).

THEOREM 3.4 —If Ha,b(Q) is a division algebra, the homogeneous space SL2(R)/Γa,b is com-pact.

REMARKS.— 1) Since SL2(R) and Γa,b are unimodular, there is an SL2(R)-invariant measureon SL2(R)/Γa,b – see corollary 2.3. When SL2(R)/Γa,b is compact, this measure is finite andΓa,b is a lattice in SL2(R).2) We have shown previously that SL2(Z) is a lattice in SL2(R), but SL2(R)/SL2(Z) is notcompact – see 3.1.3) One can show that Γa,b is always a lattice in SL2(R) and that it is cocompact if and onlyif Ha,b(Q) is a division algebra.4) We will show later on that Γa,b is indeed an arithmetic lattice in SL2(R). In order to provetheorem 3.4, we need some preliminary remarks and a lemma.

Every x∈Ha,b(R) determines two endomorphisms of the R-vector space Ha,b(R) via the leftand right multiplications: Lx(y) := xy and Rx(y) := yx. In the basis {1; i; j; k}, the matricesof Lx and Rx are

x1 x2b x3b −x4abx2 x1 x4b −x3bx3 −x4a x1 x2ax4 −x3 x2 x1

and

x1 x2a x3b −x4abx2 x1 −x4b x3bx3 x4a x1 −x2ax4 x3 x2 x1

and one verifies that det(Lx) = det(Rx) = N(x)2. Now for m∈Z \ {0}, define

Hma,b(Z) := {x∈Ha,b(Z) : N(x) = m}.

These sets are invariant under left and right multiplication by H1a,b(Z).

LEMMA 3.5.— The group H1a,b(Z) has finitely many orbits in Hm

a,b(Z) when acting either byleft or right multiplication.

Proof.— Since H1a,b(Z) and Hm

a,b(Z) are invariant under x 7→ x, it suffices to show the lemma

for the action of H1a,b(Z) by left multiplication.

Now take x∈Hma,b(Z). Observe that Ha,b(Z) and Ha,b(Z).x = Rx

(Ha,b(Z)

)are lattices in the

additive group Ha,b(R). Hence

#Ha,b(Z)/Ha,b(Z).x =Vol(Ha,b(R)/Ha,b(Z).x

)Vol(Ha,b(R)/Ha,b(Z)

) =|detRx |= N(x)2 = m2.

Now there are only finitely many subgroups of index m2∈Ha,b(Z). Let

Ha,b(Z).x1,... Ha,b(Z).xr(m)

be the set of distincts subgroups so obtained. For any x∈Hma,b(Z) there is an i between 1 and

r(m), such that

14

Ha,b(Z).x = Ha,b(Z).xi.

Setting y = xix−1 we have that y∈Ha,b(Z) and N(y) = 1, hence y∈H1

a,b(Z). This shows that

Hma,b(Z) =

r(m)⊔i=1

H1a,b(Z).xi.

�

Proof of theorem 3.4.— We have to show that H1a,b(R)/H1

a,b(Z) is compact. Let g∈H1a,b(R),

and Lg the left multiplication by g. Then Λ := Lg(Ha,b(Z)

)is a lattice in Ha,b(R). and

Vol(Ha,b(R)/Λ) =|det(Lg) |= N(g)2 = 1.

It follows from Minkowski’s lemma (proposition 3.1) that the closed convex balanced set

V = {g∈Ha,b(R) : |yi |≤ 1, 1 ≤ i ≤ 4},whose volume is 24, contains a nonzero element of λ. In other words, there exists x∈Ha,b(Z) \{0} with gx ∈ V . Since Ha,b(Q) is a division algebra, it folows that N(x) is nonzero. Inparticular

gx∈V ∩Ha,b(R)×.

Observe that N(gx) = N(x) = m and |m |=|N(x) |=|N(gx) |≤ (a+ 1)(b+ 1) since |(gx)i |≤ 1for each coordinate of gx. The set

Vm = {x∈Ha,b(R) :|xi |≤ 1, N(x) = m}is a compact set contained in Ha,b(R)× and it follows from the above discussion that gx is inVm. Applying lemma 3.5, we have x = x′xi for some x′∈H1

a,b(Z), 1 ≤ i ≤ r(m), so that

gx′ ∈ Vmx−1i ⊂

⋃1≤|m|≤(a+1)(b+1)

1≤i≤r(m)

Vmx−1i .

Let C := H1a,b(R) ∩

⋃1≤|m|≤(a+1)(b+1)

1≤i≤r(m)

Vmx−1i . This is a compact subset of H1

a,b(R) and we have

finally shown that for any g∈H1a,b(R) there exists x′∈H1

a,b(Z) such that gx is in C. �

Finally we show that Γa,b is an arithmetic lattice in SL2(R).

For every x∈H1a,b(C), let lx be the matrix of the endomorphism Lx of H1

a,b(C) w.r.t. the basis{1; i; j; k}. We obtain an algebra homomorphism

l : H1a,b(C)→M4,4(C)

x 7→ lx

which is injective. Likewise, let rx be the matrix of Rx in the basis {1; i; j; k}. One verifiesthat {ri; rj ; rk} is in M4,4(Z). Here is a characterisation of the image of l.

LEMMA 3.6.— l(Ha,b(C)

)= {A∈M4,4(C) : Ari = riA,Arj = rjA,Ark = rkA}.

Proof.— It is clear that l(Ha,b(C)

)satisfies these properties. Conversely, if A ∈ M4,4(C)

commutes with ri, rj , rk then it commutes with rx for all x∈Ha,b(C). Hence

A(y) = A(1.y) = Ary(1) = ryA(1) = A(1).y = lA(1)(y).�

15

Next we characterize l(H1a,b(C)

). For every matrix A∈M4,4(C), let

CA(λ) := λ4 − λ3c1(A) + λ2c2(A)− λc3(A)− c4(A).

Then the ci(A)’s are polynomials in the matrix entries of A with coefficints in Z. Now

clx(λ) = det(λ− lx) = N(λ− lx)2

= λ4 − 2Tr(x)λ3 + [2N(x) + Tr(x)2]λ2 − 2Tr(x)N(x)λ+N(x)2,

where Tr(x) = x+ x. Hence 2N(x) = c2(lx)− c1(lx)2.

Now set G := {A∈GL4(C) : A commutes with ri, rj , rk andc2(A)− c1(A)2 = 2}. Then G is aconnected algebraic subgroup of GL4(C) defined over Q, isomorphic as a Lie group to H1

a,b(C)

via l−1. Composing with l−1 we get a Lie group isomorphism

p : GR → SL2(R)

such that p(G(Z)

)= Γa,b.

Observe that in this example GR is connected and p is injective.

3.C Restriction of scalars and arithmetic lattices

Throughout this section, G will denote an algebraic group.

DEFINITION.— The algebraic group G is called almost simple if the adjoint representation Adis irreducible. If G is defined over k, it is called almost k-simple if Adkis irreducible.

Here is now a connection between algebraic and Lie groups.

PROPOSITION 3.7.— Let H be a connected semisimple Lie group with trivial center. Thereexists an algebraic group G < GLn(C) defined over Q such that, as Lie groups, H and G◦Rare isomorphic.

Proof.— Let h be the Lie algebra of H and Ad be its adjoint representation. The AdH =(Auth)◦, where Auth is the group of automorphisms of the real Lie algebra h. Since H hastrivial center, it is isomorphic to AdH, so that H ∼= (Auth)◦. Now let X1,... Xn be a basis ofh. The coordinates of the vectors [Xi, Xj ] in this basis

[Xi, Xj ] =n∑k=1

akijXk

are the structure constants of h. Via this basis, we identify h with Rn. The group

G := {g∈GLn(C) : [gXi, gXj ] = g[Xi, Xj ], 1 ≤ i, j ≤ n}

is an algebraic group, defined by quadratic equations with coefficients in the field Q({akij}) ⊂ R.

We have obviously that GR = Auth, hence H ∼= G◦R. According to a Chevalley’s theorem, anyreal semisimple algebra has a basis whose structure constants are in Q. Using such a basis,one obtains G defined over Q. �

16

Restriction of scalars.— In this section K = C, k is a finite extension of Q of degree d and Ois the ring of integers of k.

We are going to describe an operation which to any algebraic group G defined over k associatesan algebraic group RK/QG defined over Q and a group homomorphism

j : G(k)∼−→ RK/QG(Q)

such that j(G(O)

) ∼= RK/QG(Z). We describe this operation in the more general context of

affine varieties V ⊂ CN . We first recall a few basic facts from Galois theory.

a) There is a basis α1,... αd of k over Q such that O =d⊕i=1

Zαi.

b) There are d distinct field embeddings σi : k → C. They are linearly independant in theC-vector space of all maps k → C. In particular, the matrix [σi(αj)]i,j is invertible. We setα1 := id.

c) Q = {x∈k : σi(x) = x, 1 ≤ i ≤ d}.

d) Any field imbedding σ : k → C can be extended to an automorphism of C.

First we describe restriction of scalars for the case of the affine variety CN defined over k.Every field imbedding σi : k → C gives rise to a Q-linear map

σNi : kN → CN

w = (w1, ...wN ) 7→ (σiw1, ...σiwN )

from which one deduces a Q-linear map

ι : kN →Md,N (C)

w = (w1, ...wN ) 7→

σ1(w)...

σN (w)

Explicitly

w 7→

σ1w1 ... σ1wN... ... ...

σdw1 ... σdwN

It is plain that ι identifies the Q-vector space kN with ι(kN ), a Q-subspace of Md,N (C).Expanding wk∈k in the Q-basis α1,... αd

wk =d∑j=1

wjkαj , wjk ∈ Q,

one gets that the k-th column of the above matrix is σ1wk...

σdwk

= [σi(αj)]i,j

w1k

...wdk

.

Hence if T : Md,N (C)→Md,N (C) denotes the C-linear map obtained from left multiplicationby [σi(αj)]i,j , we have

17

ι(kN ) = T(Md,N (Q)

).

Therefore j := T−1ι identifies the k-points of the k-variety CN with the Q-points of theQ-variety Md,N (C). It follows from a) that j(ON ) = Md,N (Z).

Now let V ⊂ CN be an affine variety defined over k and I = I(V ) ⊂ C[X1, ...XN ] its definingideal. For σ∈AutC and p∈C[X1, ...XN ], we denote by pσ the polynomial whose coefficientsare obtained by applying σ to those of p. This way AutC acts by Q-algebra automorphismson C[X1, ...XN ]. For σ ∈ AutC we let V σ be the affine variety corresponding to the idealIσ = {pσ}p∈I . Since V is defined over k, V σ is defined over σ(k). Identifying (CN )d withMd,N (C), we consider ∏

V σi = V σ1 × ...× V σd

as an affine subvariety of Md,N (C). The variety∏V σi can be described as the set of common

zeroes of the polynomial mappings

p : Md,N (C)→Md,N (C)X11 ... X1N

... ... ...Xd1 ... XdN

7→ pσ1(X11, ..., X1N ) 0 ... 0

... ... ... ...pσd(Xd1, ..., XdN ) 0 ... 0

where p∈Ik(V )(:= I(V )∩ k[X1, ...XN ]). Therefore the affine variety T−1(

∏V σi) ⊂Md,N (C)

is the set of common zeroes of the polynomial mappings p := T−1 ◦ p ◦T , p∈Ik(V ). The pointof this construction is that the polynomial mappings p are with coefficients in Q. Indeed

p(Md,N (Q)

)= T−1p

(ι(kN )

)and if

σ1(w)...

σd(w)

∈ ι(kN ), w∈kN , then

p

σ1(w)...

σd(w)

=

σ1.p(w) 0 ... 0... ... ... ...

σd.p(w) 0 ... 0

=

σ1.p(w) σ1.0 ... σ1.0... ... ... ...

σd.p(w) σd.0 ... σd.0

so that T−1p .

σ1(w)...

σd(w)

is in Md,N (Q). Hence p(Md,N (Q)

)⊂ Md,N (Q), from which one

deduces easily that p has coefficients in Q. Hence the affine variety

RK/QV := T−1(∏V σi) ⊂Md,N (C)

is defined over Q. Moreover the map j = T−1ι

V (k)→∏V σi → RK/QV (Q)

w 7→

σ1(w)...

σd(w)

7→ T−1

σ1(w)...

σd(w)

identifies the k-points of V with the Q-points of RK/QV . Besides j

(V (O)

)= RK/QV (Z).

Applying this construction to an algebraic group G defined over k, one obtains

18

PROPOSITION 3.8.— k is a finite extension of Q of degree d and O is the ring of integers ofk. Let σi : k → C be the d distinct field imbeddings of k ∈C, and L be the smallest fieldcontaining the σi(k)’s. Consider the group homomorphism

ι : G→d∏i=1

Gσi

g 7→ (σ1(g), ...σd(g).

Then the L-algebraic groupd∏i=1

Gσi is isomorphic over L to a Q-algebraic group RK/QG. This

isomorphism identifies ι(G(k)

)with RK/QG(Q) and ι

(G(O)

)with RK/QG(Z). �

Restriction of scalars is most useful when combined with Borel-Harish Chandra’s theorem. Italso serves to illustrate the definition of arithmetic lattices.

EXAMPLE. —k = Q(√

2), O = Z[√

2], σ = id and σ2(x +√

2y) = x −√

2y. Set σ = σ2. LetG = SO(F ), where F : C → C is the quadratic form F (x1, x2, x3) = x2

1 + x22 −√

2x23. G is

defined over Q(√

2). Observe that k ∪ σ(k) ⊂ R. According to proposition 3.8, RK/QG is

isomorphic over R to the R-group G×Gσ. Hence RK/QGR∼−→ GR×Gσ(R), and under this

isomorphism RK/QG(Z) is sent to the subgroup

Γ := {(g, σ.g) : g∈SO(F,Z[√

2])}.

Observe that GR is isomorphic to SO2,1(R) and Gσ(R) is compact since Fσ is positive definite.It follows from Borel-Harish-Chandra that RK/QG(Z) is a lattice in RK/QGR, and hence Γis a lattice in SO(F,R) × SO(Fσ,R). Since SO(Fσ,R) is compact the projection of Γ toSO(F,R) is a lattice in this group. Hence SO(F,Z[

√2]) is a lattice in SO(F,R).

4. AMENABLE GROUPS

DEFINITION.— A Frechet space is a topological vector space V such that

(i) V is Hausdorff;(ii) the topology of V is generated by a family of seminorms ‖ − ‖α, α∈I.

EXAMPLE.— Let B be a Banach space, B∗ its dual and B∗σ the topological vector space definedby the weak-∗-topology on B∗. Then V is a Frechet space, the family of seminorms being

‖λ‖v=|λ(v) |, v∈B, λ∈B∗.

We will frequently use Banach-Alaoglu’s theorem which asserts that the unit ball in B∗

{λ∈B∗ :‖λ‖≤ 1}

is a compact subset of V = B∗σ.

DEFINITION.— A topological group L is amenable if for any continuous linear action

L× E → E

of L on a Frechet space E, and for any convex compact nonvoid L-invariant set I ⊂ E, thereis an L-fixed point in I. A group L is amenable if as a discrete group it is amenable.

19

PROPOSITION 4.1.— An abelian group is amenable.

Proof.— Let A × E → E and I ⊂ E be as in the definition above. For T ∈A, denote by ET

the closed subspace of E consisting of all T -fixed points and set IT := I ∩ ET .

For y∈I and n an integer, the convex combination

Cn(y) :=1

n+ 1[y + Ty + ...+ Tny]

is in I. For a seminorm ‖ − ‖α we have

‖ T(Cn(y)

)− Cn(y) ‖α ≤ ‖ 1

n+ 1(Tn+1y − y) ‖α

≤ 2

n+ 1maxx∈I‖ x ‖α

Hence any accumulation point of the sequence(Cn(y)

)N∈N is T -invariant: IT 6= ∅.

The group A is abelian and hence acts on the Frechet space ET , leaving the convex compactnonvoid set IT invariant. In particular, we have for all S∈A

(IT )S = IT ∩ IS 6= ∅.

By induction we get⋂T∈F

IT 6= ∅ for any finite subset F ∈A, and hence⋂T∈A

IT 6= ∅ since I is

compact. �

PROPOSITION 4.2.— An extension R of topological groups 1 → A → R → B → 1 with A andB amenable, is amenable.

Proof.— Let R × E → E and I ⊂ E be as in the definition of amenability. The spaceEA of A-fixed points is a Frechet space on which B = R/A acts. Since A is amenable,IA = EA ∩ I 6= ∅. Since I is R-invariant, IA ⊂ EA is B-invariant. Now B being amenable,we have IR = (IA)R 6= ∅. �

COROLLARY 4.3.— A solvable group is amenable. �

PROPOSITION 4.4.— A compact group is amenable.

Proof.— Let K be a compact group, α a Haar measure on K and K × E → E, I ⊂ E be as

in the definition of amenability. For v∈I,

∫K

k.v dα(k) is a K-fixed point in I. �

COROLLARY 4.5.— An extension L of topological groups 1→ R→ L→ K → 1 with R solvableand K compact, is amenable. �

COROLLARY 4.6.— Let G be a connected semisimple algebraic group defined over R and Pbe a minimal parabolic subgroup of G defined over R. The topological group P = P(R) isamenable.

Proof.— P is the semidirect product P = Z(S)nU where U is the unipotent radical of P andZ(S) is the centralizer of a maximal R-split torus S. Besides, Z(S) = M · S where M is thebiggest connected anisotropic subgroup of Z(S) and M ∩ S is finite. The subgroup B = S · Uis a solvable normal subgroup of P . Furthermore P/B is compact since so is M . The resultfollows from corollary 4.5. �

20

EXAMPLE.— G = SO(F ), the special orthogonal group of a quadratic form F : Rn → R ofindex p. In a suitable basis {e1, ... en}, F has matrix J

F0

J

where J := [δi+j,p+1] is a p× p matrix and F0 : Rn−2p → R is positive definite. The stabilizerP∈SO(F ) of the flag (e1) ⊂ (e1, e2) ⊂ ... ⊂ (e1, ...ep) is a minimal parabolic subgroup definedover R. An element of this subgroup has the formA11 A12 A13

0 A21 A22

0 0 A33

where A11 and A33 are upper triangular matrices and A33 = J tA−1

11 J , A21 ∈ SO(F0). Asmaximal R-split torus, one may choose

S = {

λ1

...λp

In−2p

λ−1p

...λ−1

1

: λi 6= 0}

whose centralizer is SO(F0). The unipotent radical U of P is the set of matricesA11 A12 A13

0 In−2p A22

0 0 A33

where A11 and A33 are unipotent. One has the decomposition P = SO(F0).S.U , and thecompact group SO(F0) is the quotient of P by the normal solvable subgroup S.U .

A remarkable property shared by amenable groups is the following.

PROPOSITION 4.7.— Let L×X → X be a continuous action of a topological amenable groupL on a compact space X. Then, there exists an L-invariant probability measure on X.

Proof.— The group L acts continuously on C(X), the Banach space of continuous functionsf with norm

‖f ‖sup:= supx∈X|f(x) |.

The dual of C(X) is M(X), the Banach space of bounded measures on X, with norm

‖µ‖mass:= sup06=f∈C(X)

〈µ, f〉‖f ‖sup

.

Then L acts continuously on the Frechet space M(X)σ (weak topology), and leaves invariantthe nonvoid compact convex subset M1(X) consisting of probability measures. The proposi-tion then follows from the assumption that L is amenable. �

21

We use now proposition 4.7 to prove that F2, the free group on two generators, is not amenable.We realize F2 as a subgroup of SL2(R) and show that on P1R there is no F2-invariant prob-ability measure.

(1) Let η, γ∈SL2(R) be hyperbolic elements, and η± and γ± be the corresponding attractingand repelling fixed points. Assume that

{γ+; γ−} ∩ {η+; η−} 6= ∅.

Claim: there is an integer n such that the subgroup of SL2(R) generated by ηn and γn is free.This is an easy consequence of Ping-Pong lemma.

Set a := γn and b := ηn.

(2) Let µ be a probability measure on P1R which is invariant under 〈a〉, the subgroup generatedby a. For any closed neighborhood V of γ−, we have⋂

k≥1

ak(P1R \ V

)= {γ+},

hence µ(γ+) = limn→∞

µ(a.(P1R \ V )

)= µ

(P1R \ V

), which shows that supp(µ) is contained in

{γ+} ∪ {γ−}. Assuming that µ is 〈a, b〉-invariant, we get

supp(µ) ⊂ ({γ+} ∪ {γ−}) ∩ ({η+} ∪ {η−}),

which contradicts the assumption that µ is a probability measure.

5. THE FURSTENBERG MAP

Let G be a Lie group and Γ < G be a lattice. Let

ρ : Γ→ GL(Vk)

be a linear representation of Γ∈Vk, a finite dimensional vector space over k. Here k is R, Cor a finite extension of Qp. The projective space PVk is a compact space on which Γ acts viathe representation ρ:

Γ× PVk → PVk(γ, x) 7→ ρ(γ)(x).

The group Γ acts by isometries on C(PVk), the Banach space of continuous functions withnorm

‖f ‖sup:= supx∈PVk

|f(x) |,

via

ρ(γ)∗f(x) := f(ρ(γ)−1x).

The dual of C(PVk) is M(PVk), the Banach space of bounded measures on PVk, with norm

‖µ‖mass:= sup06=f∈C(PVk)

〈µ, f〉‖f ‖sup

.

Then Γ acts on M(PVk) via

22

ρ(γ)∗µ(f) :=

∫PVk

f(ρ(γ).x) dµ(x),

and the action Γ×M(PVk)→M(PVk) is continuous for the weak topology onM(PVk). Thismay be verified directly. It also follows from the following general fact we will use later.

Let L be a topological group and L×B → B a continuous action on a Banach space B. ThenL acts by isometries on B∗ the dual of B and the action

L×B∗σ → B∗σ

of L on B∗σ, the dual endowed with the weak topology, is continuous.

THEOREM 5.1.— Let P < G be a closed amenable subgroup. There exists a measurable map

ψ : G/P →M1(PVk)

which is Γ-equivariant.

REMARKS. —1) Let Y be a topological space. A map ψ : G/P → Y is measurable if ψ, seenas a map G → Y , is measurable (i.e., for every Borel set B ⊂ Y , ψ−1(B) ⊂ G is measurablew.r.t. Haar measure).2)M(PVk) is endowed with the Borel structure deduced from the weak topology. One verifiesthat it coincides with the Borel structure defined by the norm topology.

Proof.— Let α be the G-invariant probability measure on Γ \G. We define

F := L1Γ

(G,C(PVk)

)to be the Banach space of all Γ-equivariant measurable maps f : G→ C(PVk) such that

‖f ‖sup,1:=

∫Γ\G‖f(g)‖sup dα(g) <∞.

The dual of F is the Banach space

E := L∞Γ(G,M(PVk)

)of all Γ-equivariant measurable maps m : G→M(PVk) such that

‖m‖mass,∞:= ess supg∈G‖m(g)‖mass<∞.

The pairing

F × E → R

(f,m) 7→∫

Γ\G〈f(g),m(g)〉 dα(g)

realizes the duality.

The formula f∗h(g) := f(gh) defines a continuous action of G on F by isometries. ThereforeG acts continuously on Eσ, Eσ being the dual of F with weak topology, via the formulam∗h(g) := m(gh), m∈E. Consider

I = {m∈E : m(g)∈M1(PVk) for almost every g∈G}.One verifies that I is G-invariant, convex, compact and non void. The last assertion followsfrom the fact that Γ is a discrete subgroup of G. Since P is amenable, there is a P -fixed pointin I. �

23

Connection with Teichmuller theory. —The group PSL2(C) acts on P1C in the obvious way:

PSL2(C)× P1C→ P1C

(

(a bc d

), z) 7→ az + b

cz + d,

and PSL2(R) leaves H2 := {z∈C : Iz > 0} ⊂ P1C invariant. Observe that the boundary ofH2 is P1R. Now, PSL2(R) is the group of orientation preserving isometries of the hyperbolicplane (

H2, ds2 =dx2 + dy2

y2

).

Let Γ, Γ′ be lattices in PSL2(R) such that

(1) The surfaces S := Γ \H2 and S := Γ′ \H2 are compact(equivalently, Γ \ PSL2(R) and Γ′ \ PSL2(R) are compact);

(2) Γ and Γ′ act without fixed point on H2;

(3) S and S′ are homeomorphic.

A homeomorphism f : S∼−→ S′ lifts to a homeomorphism F : H2 ∼−→ H2 which is equivariant

w.r.t. an isomorphism θ : Γ∼−→ Γ′, that is

F (γ.z) = θ(γ)F (z) ∀z∈H2 ∀γ∈Γ.

Now F is easily seen to be a quasi-isometry of H2, and therefore extends via Mostow’s con-struction to a quasi-symmetric homeomorphism

ϕ : P1Rq.s.−→ P1R

which is again Γ-equivariant.

On the other hand, we have associated to the representation

θ : Γ→ Γ′ < PSL2(R)

a Furstenberg map

ψ : P1R→M1(P1R

)which is Γ-equivariant measurable. It can be shown that in this situation, ψ takes its valuesin the Dirac measures on P1R, and in fact that ψ(x) = δϕ(x) ∀x∈P1R.

6. THE ZARISKI SUPPORT MAP

Let k be a local field, i.e., R, C or a finite extension of Qp. Let Vk be finite dimensionalk-vector space, V := Vk ⊗K where K is an algebraic closure of k and Vark(PV ) be the spaceof projective subvarieties of PV defined over k – see below. In this chapter, we study the map

M1(PVk)suppZ−→ Vark(PV )

µ 7→ supp(µ)Z

24

which to a probability measure µ associates the Zariski closure of its support. We will provethat for a natural topology on Vark(PV ), this map is Borel.

The map suppZ is the composition of the following two maps.

(a)M1(PVk)

supp−→ PF (PVk)µ 7→ supp(µ)

the map that associates to every measure µ its support supp(µ). Here PF (PV ) is the space ofclosed subspaces of PVk.

(b)PF (PVk)

ClZ−→ Vark(PV )

F 7→ FZ

the map which to every closed subset F ⊂ PV associates its Zariski closure.

6.A supp :M1(PVk)→ PF (PVk)

We endow the space PF (PVk) of closed subspaces of PVk with a topology which has thefollowing two equivalent descriptions.

(1) Let d be a metric on PVk which generates its topology and set for F1, F2∈PF (PVk):

D(F1, F2) := max{d(f1, F2), d(f2, F1) : f1∈F1, f2∈F2}.

One verifies that D is a metric on PF (PVk): the Hausdorff metric. One can show that PF (PVk)is a compact metric space.

(2) For any open set U ⊂ PVk, define

TU := {F ∈PF (PVk) : F ∩ U 6= ∅}; IU := {F ∈PF (PVk) : F ⊂ U}.

Taking unions and finite intersections of TU ’s and IU ’s provides a topology on PF (PVk).

It is clear that open sets for the topology (2) are open for the topology (1). Conversely, letF ∈PF (PVk) and B(F, ε) an open ε-ball with center F in the Hausdorff metric. Cover F byfinitely many ε/2-balls:

F ⊂n⋃i=1

Biε/2

and let Fε/2 be the open ε/2-neighborhood of F . One verifies that

IFε/2 ∩n⋂i=1

TBiε/2⊂ B(F, ε).

LEMMA 6.1.— supp is a PGL(Vk)-equivariant Borel map.

Proof.— The map is clearly PGL(Vk)-equivariant.

(a) We have for U ⊂ PVk open:

{µ : supp(µ)∈TU} = {µ : supp(µ) ∩ U 6= ∅} = {µ : µ(U) > 0}.

It suffices to verify that the evaluation map

M1(PVk)→ Rµ 7→ µ(U)

25

is Borel. Indeed, let {fn : PVk → [0; 1]}n be a sequence of continuous functions convergingpointwise to χU , the characteristic function of U . By Lebesgue’s dominated convergencetheorem, we have for every µ∈M1(PVk): µ(U) = lim

n→∞µ(fn). Therefore the above evaluation

map is pointwise limit of the sequence of continuous maps

M1(PVk)→ Rµ 7→ µ(fn),

hence is Borel.

(b) The open set U ⊂ PVk can be written as a countable union of closed sets U =⋃n≥1

Fn.

So {µ : supp(µ)⊂U} =⋃n≥1

{µ : supp(µ)⊂Fn} =⋃n≥1

{µ : µ(PVk \ Fn) = 0} is Borel. �

6.B ClZ : PF (PVk)→ Vark(PV )

Recall we are given Vk and V := Vk⊗K, where K is an algebraic closure of k. Denote by K[V ]the space of polynomials on V , and by k[V ] the space of polynomials on V with coefficientson k. Let π : V \ {0} → PV be the canonical projection.

DEFINITION.— A projective subvariety is a set S ⊂ PV such that π−1(S) ∪ {0} is the set ofcommon zeroes of an ideal of polynomials in K[V ].

We denote by I(S) the set of polynomials vanishing on π−1(S) ∪ {0}. Let K[V ]d be the spaceof homogeneous polynomials of degree d on V . Then

K[V ] =∞⊕d=0

K[V ]d

is a graded algebra. The defining ideal of a projective variety has the fundamental property:

I(S) =∞⊕d=0

I(S)d,

where I(S)d := I(S) ∩K[V ]d, i.e., I(S) contains all homogeneous components of it elements.Indeed, let P ∈I(S) and x∈π−1(S)∪ {0}. Then P vanishes on λx for all λ∈K. If P =

∑d Pd

is the decomposition of P into its homogeneous components, we have∑d λ

dPd(x) = 0, for all λ∈K.

Since K is infinite, this implies Pd(x) = 0. �

DEFINITION.— A projective variety S ⊂ PV is defined over k if I(S) := I(S) ∩ k[V ] generatesI(S) over K.

Obviously I(S) =∞⊕d=0

Id(S) where Id(S) := I(S) ∩ k[V ]d.

One verifies that intersections and finite unions of projective varieties are projective varieties.They form a family of closed sets of a topology on PV called Zariski topology.

The injection Vk ↪→ V induces an injection PVk ↪→ PV .

26

LEMMA 6.2.— Let F ⊂ PVk be a subset. Then the Zariski closure FZ ⊂ PV is defined over k.

Proof.— Denote by π : V \ {0} → PV and by πk : Vk \ {0} → PVk the canonical projections.By definition

I(FZ) = {P ∈K[V ] : P |π−1(F )= 0}

and hence

Id(FZ

) = {P ∈K[V ]d : P |π−1(F )= 0} = {P ∈K[V ]d : P |π−1k

(F )= 0}.

Let r = dimKK[V ]d−dimKI[V ]K. By linear algebra there exists x1, ... xr∈π−1k (F ) ⊂ Vk \{0}

such that

{P ∈K[V ]d : P |π−1k

(F )= 0} = {P ∈K[V ]d : P (xi) = 0, 1 ≤ i ≤ r}.

Now {P (xi)}1≤i≤r is a linear system of r equations with coefficients in k, the unknown beingthe coefficients of P . By linear algebra, the K-vector space of solutions is generated by the

k-vector space of solutions with coordinates in k. Hence Id(FZ

) = Id(FZ

)⊗K, and hence FZ

is defined over k. �

Let Vark(PV ) be the set of projective varieties of PV defined over k. Recall that

I(S) = I(S) ∩ k[V ] and I(S) =⊕d

Id(S),

for S ∈Vark(PV ), where Id(S) = {P ∈K[V ]d : P vanishes on S} (for a homogeneous polyno-mial, vanishing on a set in PV makes sense).

NOTATION. —For a finite dimensional k-vector space Wk, we set

Gr(Wk) :=d⊔l=0

Grl(Wk),

where Grl(Wk) is the Grassmannian of l-planes in Wk. This is a compact topological space.Via the injection

Vark(PV )→∞∏d=0

Gr(k[V ]d)

S 7→(Id(S)

)d∈N

we consider Vark(PV ) as a subset of the compact topological space

∞∏d=0

Gr(k[V ]d). We endow

Vark(PV ) with the induced topology. In virtue of lemma 6.2, we may consider the map

ClZ : PF (PVk) → Vark(PV )

F 7→ FZ

which will turn out to be Borel. We need the following description of Id(FZ

).

The Plucker imbedding in degree d is the map

Pld : PVk → Pk[V ]∗d

27

defined as follows. Every vector v ∈ Vk gives by evaluation a linear form on k[V ]d, denoted

by Pld(v). The map v 7→ Pld(v) is homogeneous of degree d, and so defines Pld, which is ahomeomorphism onto its image.

For each k-vector space Wk, we also have a polarity map Pol : Gr(W ∗k )→ Gr(Wk), which is ahomeomorphism.

Now the map Pld : PVk → Pk[V ]∗d induces a continuous map

Pld : PF (PVk) → PF(Pk[V ]∗d

)F 7→ Pld(F )

Let [−]lin : PF(Pk[V ]∗d

)→ Gr(k[V ]∗d) denote the map which to a closed subset F of Pk[V ]∗d

associates its linear space, seen as an element of Gr(k[V ]∗d). We have

Id(FZ

) = Pol(

[Pld(F )]lin

).

This is a straightforward verification.

LEMMA 6.3.— Let Wk be a finite dimensional k-vector space.

(i) The fibers of the map

dim : PF (PWk) → NF 7→ dim[F ]lin

are locally closed.(ii) For any interger l, the restriction to dim−1(l) of the map

PF (PWk) → Gr(Wk)F 7→ [F ]lin

is continuous.

Proof.— Let F , F ′ ∈ PF (PWk) such that D(F, F ′) < ε. Set l := dim[F ]lin. Take x1,... xl∈Fsuch that

[F ]lin = [{x1,... xl}]lin.

Let y1,... yl∈F ′ such that d(xi, yi) < ε for 1 ≤ i ≤ l. For ε sufficiently small, dim[{y1, ...yl}]lin =l, hence dim[F ′]lin ≥ dim[F ]lin, which shows that dim is lower semicontinuous and proves (i).If now dim[F ′]lin = dim[F ]lin, then [F ′]lin is near [F ]lin∈Gr(Wk), which proves (ii). �

PROPOSITION 6.4.— The map ClZ : PF (PVk)→ Vark(PV ) is Borel, PGL(Vk)-equivariant.

Proof.— PGL(Vk)-equivariance is clear. By definition of the topology on Vark(PV ), we haveto verify that for all integer d, the map

PF (PVk) → Grk[V ]d

F 7→ Id(FZ

)

is Borel. But this map is the composition

PF (PVk)Pld−→ PF

(Pk[V ]∗d

) [−]lin−→ Gr(k[V ]∗d)Pol−→ Gr(k[V ]d).

Now, Pld and Pol are continuous, and it follows from lemma 6.3 that [−]lin is Borel. �

28

6.C Combining the above results, we finally get:

THEOREM 6.5.— The Zariski support map suppZ : M1(PVk) → Vark(PV ) is Borel andPGL(Vk)-equivariant. �

7. ERGODIC ACTIONS AND BOREL STRUCTURES

7.A Ergodic actions

Let X be a locally compact and countable at infinity topological space.

DEFINITIONS.— (i) µ, ν positive Radon measures on X are equivalent if the family of µ-negligible sets coincides with the family of ν-negligible ones.

(ii) The class of a positive Radon measure µ is the set of positive Radon measures equivalentto µ.

(iii) Let R be a group acting on X by homeomorphisms. The class of µ is R-invariant if forall r∈R, r∗µ and µ are equivalent.

(iv) In the previous situation, the action of R is ergodic if for any R-invariant measurable setE ⊂ X, either E or X \ E is µ-negligible.

For our purpose, we study actions on measure spaces (X,µ) of the following type: X = G/H,where G is a Lie group and H is a closed subgroup of G. Then G/H is a C∞-manifold on whichG acts by diffeomorphisms. All Riemannian volumes (constructed via metrics) on G/H are inthe same measure class, and G preserves this class, called the canonical class. For example, theHaar measure of G is in the canonical class. Besides, if G/H carries a G-invariant measure,this measure is in the canonical class. The projection p : G → G/H is a fibration. From thisand Fubini’s theorem follows that, for all E ⊂ G/H measurable

E is of measure 0 ⇐⇒ p−1(E) is of measure 0.

From this we get immediately

LEMMA 7.1.— Let R, H be closed subgroups of G. The R-action on G/H is ergodic if andonly if the H-action on G/R is ergodic. �

We now turn to the main result of this chapter.

THEOREM 7.2.— Let G be a simple connected Lie group and t be an element of G such thatAdt is diagonalizable with positive real eigenvalues, not all = 1. Let A = 〈t〉. Let Γ be alattice. The action of A on Γ \G is ergodic.

From lemma 7.1 and theorem 7.2, we get

COROLLARY 7.3.— Let G, Γ, t∈G be as above, and let R < G be a closed subgroup containingt. The action of Γ on G/R is ergodic. �

29

EXAMPLE.— G = PSL2(R); Γ < G is a lattice;

t =

(λ 00 λ−1

); P = {

(∗ ∗0 ∗

)∈ PSL2(R)}.

Then 〈t〉 and hence P acts ergodically on Γ \G and therefore Γ acts ergodically on G/P , thatis P1R.

Theorem 7.2 follows from a theorem on unitary representations of G.

DEFINITION.— A unitary representation of G in a Hilbert space H is a homomorphism π :G→ U(H) of G into the group of unitary opertaors of H such that the corresponding action

G×H → H(g, v) 7→ π(g)v

is continuous.

EXAMPLE. —Let α be the G-invariant probability measure on Γ \ G and denote L2(Γ \ G,α)by H. Then

π(g)f(x) := f(xg), f ∈H, g∈Gdefines a unitary representation of G∈H.

THEOREM 7.4.— Let G, t, A be as above. Let (π,H) be a unitary representation of G. AnyA-invariant vector is G-invariant.

Proof.— Let g = Lie G and g = g+ ⊕ g0 ⊕ g−, where

g− := {X∈g : limn→∞

Ad(t)nX = 0}g+ := {X∈g : lim

n→−∞Ad(t)nX = 0}

g0 := {X∈g : Ad(t)X = X}Let HA be the closed subspace of H consisting of all A-invariant vectors. For v ∈ HA andg = expX, X∈g, we have:

‖π(g)v − v‖ = ‖π(g)π(t−n)v − π(t−n)v‖= ‖π(tn)π(g)π(t−n)v − v‖= ‖π

(exp(AdtnX)

)v − v‖ .

Hence for X∈g−, we have

‖π(g)v − v‖= limn→∞

‖π(exp(AdtnX)

)v − v‖= 0,

and for X∈g+:

‖π(g)v − v‖= limn→−∞

‖π(exp(AdtnX)

)v − v‖= 0.

In both cases π(g)v = v.

Since every element of g0 commutes with A, it leaves HA invariant. So the group generatedby expg−, expg0 and expg+ leaves HA invariant. Since this group contains a neighborhoodof e and G is connected, this group is G.

Hence, we get a representation

G→ U(HA).

The kernel of this homomorphism contains expg− and expg+, and hence is of positive dimen-

sion. Since G is simple and connected, this kernel is G, and therefore HG = HA. �

30

Proof of theorem 7.2.— Let α be the G-invariant probability measure on Γ \ G and H :=L2(Γ \G,α), with

π(g)f(x) := f(xg), f ∈H, g∈G

Since α is G-invariant, (π,H) is a unitary representation of G. Let E ⊂ Γ \ G be an A-invariant measurable set. Assume α(E) > 0. Let χE denote the characteristic function of E.Then χE is a nonzero vector in H and is A-invariant. Hence it is G-invariant – see theorem7.4. Since G acts transitively on Γ \ G, χE coincides almost everywhere with χΓ\G. Hence

α((Γ \G)− E

)= 0. �

COROLLARY 7.5.— Let G be a simple algebraic group defined over R such that G = G◦R is notcompact. Let P be a minimal parabolic subgroup of G defined over R, and P := P ∩G. LetΓ be a lattice. The diagonal action of Γ on G/P ×G/P is ergodic.

Proof.— Denote by O the set of couples of opposite chambers in G/P × G/P : it is a singleorbit under G, identified with G/A. The complement of O is a finite union of subvarieties ofstrictly lower dimension. This complement is therefore of measure zero. Hence, ergodicity ofΓ on G/P ×G/P is equivalent to ergodicity of Γ on G/A. But this follows from corollary 7.3.�

7.B Borel structures

DEFINITION.— A Borel space is a set Y together with a σ-algebra of subsets.

Every topological space has an associated Borel structure.

Now let R, X, µ be as in the definition of ergodicity; in particular, assume that the R-actionon X is ergodic. Let Y be a Borel space and f : X → Y be an R-invariant measurable map.From the point of view of the measure µ, the R-action behaves like a transitive action. Oneexpects therefore f to be constant almost everywhere. In order for this assertion to be true,we need some additional hypothesis on the Borel space Y .

DEFINITION.— A Borel space Y is countably separated if there exists a countable family ofBorel sets separating points of Y .

A countably separated Borel space Y admits a countable coding which is Borel. More precisely,endow {0; 1} with the discrete topology, and {0; 1}N with the product topology. Now put on{0; 1}N the associated Borel structure. Let (Ai)i∈N be a family of Borel sets which separatespoints of Y . Then the map

ϕ : Y → {0; 1}Ny 7→

(χAi(y)

)i∈N

is an injective Borel map; one can show that its image is a Borel subset of {0; 1}N and that ϕis a Borel isomorphism onto it.

LEMMA 7.6.— Let R, X, µ be as in the definition of ergodicity. Assume that R acts ergodicallyon X. Let Y be a countably generated Borel set and f : X → Y be an R-invariant measurablefunction. Then f is constant almost everywhere.

Proof.— Since X is countable at infinity, we may assume that µ(X) = 1. For any Borel subsetA ⊂ Y , f−1(A) is an R-invariant measurable set and therefore of measure 0 or 1. Therefore,

31

there exists at most one point p∈Y such that µ(f−1(p)

)= 1. We have to show that there is

one.

Let (An)n∈N be a family of Borel sets separating points of Y . Then for every p∈Y , the set

Xp(n) := f−1( ⋂

1≤i≤np∈Ai

Ai)

is of measure 1 or 0. Now we have

f−1(p) =∞⋂n=1

Xp(n)

and hence µ(f−1(p)) is the limit of the stationnary sequence(µ(Xp(n))

). Hence there is np

such that

µ(f−1(p)

)= µ

(Xp(np)

).

Clearly {Xp(np)}p∈Y is a countable family of measurable sets of X which cover X. Thereforethere is p∈Y such that µ

(Xp(np)

)> 0, and so µ

(Xp(np)

)= µ

(f−1(p)

)= 1. �

LEMMA 7.7.— Let R×X → X be an ergodic action. Assume that the topological space R \Xhas a countably separated Borel structure. Then there exists an R-orbit O ⊂ X such thatµ(X −O) = 0.

Proof.— Consider the quotient map X → R \X and apply lemma 7.6. �

This means that an ergodic action R ×X → X is either essentially transitive or the quotientBorel structure is very complicated.

It is clearly desirable to have criteria on an action R × X → X which guarantee that thequotient R \X is countably separated.

LEMMA 7.8.— Let R be a group acting by homeomorphisms on a metrisable separable spaceY . If all R-orbits are locally closed, then the topological space R\Y has a countably separatedBorel structure.

Proof.— The canonical projection p : Y → R \ Y is an open map. Indeed if O ⊂ Y is open,the set

p−1(p(O)

)=⋃r∈R

rO

is a union of open sets and therefore open. Now Y has a countable basis of open sets, andso does R \ Y . Therefore it suffices to show that any two distinct points x, y∈R \ Y can beseparated by an open set.

Assume first y 6∈ {x}. Then (R \ Y )−{x} is an open set containing y but not x. Assume nowy ∈ {x}. By assumption, {x} is open in {x} and therefore there exists V ⊂ R \ Y open with

V ∩ {x} = {x}.

Clearly V contains x but not y. �

32

PROPOSITION 7.9.— Let R be locally compact countable at infinity acting continuously on Ymetrisable separable complete. Then the R-orbits in Y are locally closed if and only if eachorbit map R/StabR(y)→ R.y is a homeomorphism onto its image.

Proof.— (⇐=) : R/StabR(y) is locally compact. Hence R.y is locally compct, and thereforelocally closed.

(=⇒) : Consider the orbit map

R/StabR(y) → R.yr 7→ r.y

By assumption, R.y is open in R.y. Let U be a compact neighborhood of e∈R, and D ⊂ Rbe a countable dense set. Hence

R =⋃d∈D

dU ,

and therefore R.y =⋃d∈D

dUy. By Baire’s category theorem there exists d ∈ D such that

dUy is a neighborhood of one of its points. Therefore Uy has the same property. Now letN be a neighborhood e ∈ R and choose U such that U · U ⊂ N , U−1 = U . Now Uy is aneighborhood of one of its points, say uy and hence Ny ⊃ u−1Uy is a neighborhood of y. Themap R/StabR(y)→ R.y is continuous, injective and open: it is a homeomorphism. �

8. THE BOUNDARY MAP. THE FIRST CASE OF DICHOTOMY

8.A Let G be a simple connected Lie group and Γ < G a lattice. Let H be a connectedalgebraic group defined over k and such that the adjoint representation

Ad : Hk → GL(Vk)

is irreducible (V = LieG).

Let π : Γ → Hk be a homomorphism such that π(Γ) is Zariski dense in H. Composing theabove arrows, we obtain a representation

Γ→ GL(Vk).

To this situation we associated – see Chapter 5 – the Furstenberg map

ϕ : G/P →M1(PVk)

which is Γ-equivariant measurable. Recall that P is a minimal parabolic subgroup of G. Wecompose ϕ with the GL(Vk)-equivariant Borel map

SuppZ :M1(PVk)→ Vark(PV )

to obtain the boundary map

Φ : G/P → Vark(PV )

We are going to analyze properties of the homomorphism π in terms of properties of theboundary map Φ.

33

THEOREM 8.1.— If Φ is essentially constant, π(Γ) < Hk is relatively compact.

The notion of essential image.— Let X be a locally compact space countable at infinity, µ bea positive Radon measure on X and Y a topological space. Let

f : X → Y

be µ-measurable map. A point y∈Y is an essential value of f if for every neighborhood V ofy

µ(f−1(V )

)> 0.

The set of essential values of f , Essv(f) is called the essential image of f . It is a closed subsetof Y . The map f is said essentially constant if Essv(f) is reduced to a point.

8.B First we need a key result due to H. Furstenberg, concerning the dynamics of the actionof PGL(Vk) on M1(PVk).

PROPOSITION 8.2.— Let µ, ν ∈M1(PVk) and {gn}n≥1 a sequence of PGL(Vk)N such that

limn→∞

(gn)∗µ = ν.

If {gn}n≥1 is not relatively compact, there exists Ak, Bk proper linear subspaces of PVk suchthat

supp(ν) ⊂ Ak ∪Bk.

Furthemore,

dimA+ dimB = dimV − 1.

Proof.— Assume that {gn}n≥1 is not relatively compact. Up to passing to a subsequence, wemay assume that lim

n→∞gn = g, where g is in PEnd(Vk) and g 6∈PGL(Vk). Furthemore, we may

assume that

limn→∞

gn.Ker(g) = Ak ∈ Gr(Vk).

Let Bk := Im(g) ⊂ PVk. For every l∈PVk \Ker(g), we have lim gn(l)∈Bk, and for l∈Ker(g),lim gn(l)∈Ak.

For f : PVk → R continuous with compact support in PVk \(Ak∪Bk), the sequence (f ◦gn)n≥1

tends to 0 pointwise. By Dominated Convergence, we have ν(f) = 0. �

COROLLARY 8.3.— Let ν∈M1(PVk) and assume that supp(ν) is not contained in the union oftwo proper linear subspaces. Then

StabPGL(Vk)(ν) := {g∈PGL(Vk) : g∗ν = ν}

is compact.

Proof.— It follows from proposition 8.2 that StabPGL(Vk)(ν) is sequentially compact and hencecompact. �

34

8.C Now we assume that Φ = SuppZ ◦ ϕ is essentially constant and that Y ∈ Vark(PV ) isits essential image. Since Φ is Γ-equivariant, the projective variety Y is π(Γ)-invariant andtherefore H-invariant since π(Γ) is Zariski dense in H. In particular, Yk ⊂ PVk is Hk-invariant.Therefore, ϕ takes its values in

MZD(Yk) := {µ∈M1(PVk) : supp(µ) ⊂ Yk and suppZ(µ) ⊂ Y }.

LEMMA 8.4.— Let µ, ν ∈MZD(Yk) and {hn}n≥1 a sequence of HNk such that

limn→∞

Ad(hn)∗µ = ν.

The sequence {hn}n≥1 is relatively compact. In particular,

(i) The Hk-orbits in MZD(Yk) are closed.(ii) For every µ∈MZD(Yk), StabHk(µ) is compact.

Proof.— Assume that {hn}n≥1 is not relatively compact and hence {Ad(hn)}n≥1 is not rela-tively compact in AdHk < PGL(Vk). There are Ak, Bk proper linear subspaces of PVk suchthat

supp(ν) ⊂ Ak ∪Bkand

dimA+ dimB = dimV − 1,

see proposition 8.2. Since supp(ν) ⊂ Ak ∪ Bk, we have Y = suppZ(ν) ⊂ A ∪ B. Therepresentation

Ad : Hk → GL(Vk)

being irreducible, we have PVk = [Yk]lin ⊂ [Ak ∪ Bk]lin. In particular, [A ∪ B]lin = PV . Itfollows by dimension that A ∩B = ∅. Therefore

U := {h∈H : hA ∩B = ∅ and A ∩ hB = ∅}

is a nonvoid Zariski open subset of H. For all h∈U we have

Y = Y ∩ hY ⊂ (A ∪B) ∩ (hA ∪ hB) = (A ∩ hA) ∪ (B ∩ hB)

Since [Y ]lin = PV , it follows by dimension that A = hA and B = hB. Therefore U is alsoZariski closed. Since H is connected, we have H = U . In particular, Ak is Hk-invariant. Thiscontradicts the assumption that Ad is irreducible. �

COROLLARY 8.5.— Orbits of the diagonal action of Hk∈MZD(Yk)×MZD(Yk) are closed.

Proof.— Follows immediately from lemma 8.4. �

Proof of theorem 8.1.— We first show that the essential image of the map

ϕ : G/P →MZD(Yk)

is reduced to a point. The map

ϕ× ϕ : G/P ×G/P →MZD(Yk)×MZD(Yk)

is measurable and equivariant for the diagonal action of Γ on the source and the diagonalaction of Hk on the target. Let

q :MZD(Yk)×MZD(Yk)→ Hk \(MZD(Yk)×MZD(Yk)

)35

be the canonical projection. The measurable map q ◦ϕ is Γ-invariant. It follows from corollary8.5 and lemma 7.8 that the Borel structure of Hk \

(MZD(Yk) × MZD(Yk)

)is countably

separated. Since the action of Γ on G/P ×G/P is ergodic – see corollary 7.5, it follows fromlemma 7.6 that q ◦ ϕ is essentially constant. Therefore, there is an Hk-orbit

O ⊂MZD(Yk)×MZD(Yk)

such that for almost every (x, y)∈G/P×G/P , (ϕ×ϕ)(x, y) is in O. Since O is closed (corollary8.5), we have:

Essv(ϕ)× Essv(ϕ) = Essv(ϕ× ϕ) ⊂ O.

Therefore O contains a point of the diagonal

∆ ⊂MZD(Yk)×MZD(Yk)

and O being an Hk-orbit, we have O ⊂ ∆. Therefore

Essv(ϕ)× Essv(ϕ) ⊂ ∆

which shows that Essv(ϕ) is reduced to a point. Let Essv(ϕ) = {µ}. Since ϕ is Γ-equivariant,π(Γ) < StabHk(µ). So π(Γ) is relatively compact. �

8.D In the case G = SL2(R), we obtain more precise results.

THEOREM 8.6.— Let Γ < SL2(R) be a lattice and let

π : Γ→ SL2(R)

be a homomorphism such that π(Γ) is Zariski dense in SL2(R). There exists a unique Γ-equivariant measurable map

ϕ : P1R→ P1R.

Proof.— Consider the boundary map

Φ : P1R→ VarR(P1C)

associated to π. We can parametrize VarR(P1C) as follows:

VarR(P1C) = {P1C} t⊔n≥1

(P1R)n/Sn,

where Sn denotes the group of permutations on n letters. By ergodicity of the Γ-action onP1R, we observe that the essential image of Φ is either {P1C} or contained in (P1R)n/Sn forsome n. If it were {P1C}, Φ would be essentially constant and hence by theorem 8.1, π(Γ)would be relatively compact. Hence π(Γ) would be contained in a conjugate of SO2(R), whichis a proper algebraic subgroup of SL2(R), contradicting the assumption that π(Γ) is Zariskidense in SL2.

To prove the theorem, it suffices to show that if

ψ : P1R→ (P1R)n/Sn (n ≥ 2)

is any Γ-equivariant map, its essential image is contained in the diagonal ∆n ⊂ (P1R)n/Sn.Assume the contrary, then by ergodicity of the Γ-action on P1R, ψ(x) 6∈ ∆n for almost everyx∈P1R. Now consider

ψ × ψ : P1R× P1R→ (P1R)n/Sn × (P1R)n/Sn.

36

This map is Γ-equivariant for the diagonal action of Γ on the source and the diagonal actionof SL2(R) on the target. One verifies that the orbits of the latter action are locally closed sothat there exists an SL2(R)-orbit

O ⊂ (P1R)n/Sn × (P1R)n/Snsuch that (ψ(x), ψ(x)) is in O for almost every (x, y) ∈ P1R × P1R. Therefore we have thefollowing:

For almost every (x, y)∈P1R× P1R

(1) ψ(x) and ψ(y) are not on the diagonal;

(2) for all γ∈Γ, (ψ(γx), ψ(y))∈O.

Take one point (x, y) satisfying these two properties. It follows from (2) that for any γ thereis an h∈SL2(R) such that

(ψ(γx), ψ(y)) = h(ψ(x), ψ(y)).

In particular, h∈Stab(ψ(y)

)and h−1π(γ)∈Stab

(ψ(x)

). Hence

π(Γ) < Stab(ψ(y)

).Stab

(ψ(x)

).

But ψ(x) and ψ(y)are not contained in the diagonal of (P1R)n/Sn, which implies that Stab(ψ(y)

)and Stab

(ψ(x)

)are contained in some conjugate of

AR = {(a 00 a−1

): a∈R×}

From this follows that π(Γ) is not Zariski dense in SL2. A contradiction. �

9. THE BOUNDARY MAP. THE SECOND CASE OF THE DICHOTOMY

Under the hypothesis of 8.1, we assume now that the boundary map

Φ : G/P → Vark(PV )

is not essentially constant. Recall that

Vark(PV ) ⊂∞∏d=0

Gr(k[V ]d

),

see Chapter 6. Let pd : Vark(PV )→ Gr(k[V ]d

)denote the canonical projection. There exists

d ≥ 1 such that

pd ◦ Φ : G/P → Gr(k[V ]d

)is not essentially constant. Besides, we have Gr

(k[V ]d

)=⊔l≥1

Grl(k[V ]d

). Since Γ acts ergodi-

cally on G/P , there exists l ≥ 1 such that for almost every x∈G/P ,

(pd ◦ Φ)(x)∈Grl(k[V ]d

).

At this point, we need the following fundamental result.

THEOREM 9.1.— Let k be a field, W an algebraic variety defined over k, H an algebraic groupdefined over k and H ×W → W an algebraic action of H on W , defined over k. Then theorbits of Hk∈Wk are locally closed.

Reference.— This is [A′C-B, theorem 6.1]. �

37

From this, it follows that the orbits of Hk∈Grl(k[V ]d

)are locally closed. It follows from the

fact that Γ operates ergodically on G/P that there exists an Hk-orbit O ⊂ Grl(k[V ]d

)such

that

(pd ◦ Φ)(x)∈O, for almost every x∈G/P .

Choose p∈O. Then StabHk(p) = StabH(p) ∩Hk. Setting L := StabH(p), we see that L is analgebraic subgroup of H defined over k, and StabHk(p) = Lk. Since all Hk-orbits in Grl

(k[V ]d

)are locally closed, the orbit map

Hk/Lk → OhLk 7→ h(p)

is a homeomorphism (proposition 7.3).

We can therefore consider pd ◦ Φ as a Γ-equivariant measurable map

ϕ : G/P → Hk/Lk.

Since ϕ is not essentially constant, L 6= H. Summarizing the results of chapter 8 and 9.1, weobtain

THEOREM 9.2.— Let G be a connected simple Lie group, Γ < G be a lattice, H a k-almostsimple connected group and

π : Γ→ Hk

a homomoprhism such that π(Γ) is Zariski dense in H. If π(Γ) is not relatively compact inHk, there exists L < H an algebraic subgroup defined over k with L 6= H, and a measurablemap

ϕ : G/P → Hk/Lk,

where P is a minimal parabolic subgroup of G. �

Recall that our goal is to extend the homomorphism π to G. Without further assumptions onΓ or G, theorem 9.2 is as far as we can go. In order to proceed further, one can make one ofthe following assumptions.

(1) CommGΓ is dense in G.

(2) G has rank ≥ 2.

Case (1) will be treated in the chapter 10, case (2) in chapter 11.

10. COMMENSURATOR SUPERRIGIDITY

Let now G be a connected simple Lie group, Γ < G a lattice, Λ a subgroup such that

Γ < Λ < CommGΓ,

H a k-almost simple connected group and

π : Λ→ Hk

a homomorphism such that π(Λ) is Zariski dense in H.

38

10.1. Our first goal is to construct a Λ-equivariant measurable map from G/P into a space ofthe form Hk/Lk. This does not follow directly from theorem 9.2 since we have not made anyassumption on the topological nature of Λ. In fact, we will eventually assume that Λ is densein G.

We consider now the family F consisting of all pairs (ϕ,M) where

(1) M is an Hk-homogeneous space of the form Hk/Lk, where L is an algebraic subgroup ofH defined over k, such that L 6= H,

(2) there exists Γ′ < Λ commensurable with Γ such that

ϕ : G/P →M

is Γ′-equivariant measurable.

THEOREM 10.1.— Assume that π(Γ) < Hk is not relatively compact. Then F 6= ∅. Furthe-more, there exists (ψ,N) ∈ F , unique up to Hk-equivariant isomorphism, such that for all(ϕ,M)∈F , there exists an Hk-equivariant map p : N →M such that the diagram

G/Pψ−→ N↘ ↓ pϕ M

commutes.

Proof.— (a) F 6= ∅: this follows from theorem 9.2 provided we can show that π(Γ) is Zariski

dense in H. Our assumption is that π(Λ) is dense in H. Let L = π(Γ)Z

be the Zariski closureof π(Γ)∈H. For all λ∈π(Λ), π(Γ)/

(π(Γ)∩ λ−1π(Γ)λ

)is finite (recall Λ < CommGΓ). Hence,

the image of π(Γ)∈H/(L ∩ λ−1Lλ) is finite. Therefore π(Γ) is contained in a Zariski closedsubset of the form

n⊔i=1

hi(λ−1Lλ),

and so is L. This shows that L/(L ∩ λ−1Lλ) is finite and hence

L◦ = (L ∩ λ−1Lλ)◦ < L◦ ∩ λ−1L◦λ.

This shows that π(Λ) normalizes L◦ and hence L◦ / H since π(Λ) is Zariski dense. But H isk-almost simple, so L◦ = H.

(b) On F , we define the following preorder: (ϕ1,M1) ≥ (ϕ2,M2) if there exists an Hk-equivariant map p : M1 →M2 such that

G/Pϕ1−→ M1

↘ ↓ pϕ2 M2

commutes.

Let F0 = F/ ∼ be the quotient of F under the equivalence relation (ϕ1,M1) ∼ (ϕ2,M2) if(ϕ1,M1) ≥ (ϕ2,M2) and (ϕ2,M2) ≥ (ϕ1,M1). Then, ≥ defines an order on F0. Let H be theset of homogeneous spaces of the form (1) above with the obvious preorder structure, and letH0 be the associated ordered set. The map

39

F : F → H(ϕ,M) 7→M

induces an increasing map of ordered spaces F : F0 → H0. The theorem will follow if we canshow that F0 has a maximal element.

First, we need the following observation: if M1 ≥M2 and M2 ≥M1, then any Hk-equivariantmap M1 → M2 is an isomorphism. Let pi : Mi → M3−i be Hk-equivariant maps. It clearlysuffices to show that q := p2◦p1 : M1 →M1 is an isomorphism. Now identify M1 with Hk/Lk.Let q(eLk) = hLk. Since q is Hk-equivariant, we have for all l∈Lk:

lhLk = lq(eLk) = q(lLk) = q(eLk) = hLk,

and hence h−1Lkh < Lk. Consider Un =

n⋂i=0

h−iLhi; {Un}n≥0 is a decreasing sequence of

algebraic subgroups of H. It stabilizes: there exists n such that Un = Un+1; and hence

h−nLkhn = (Un)k = (Un+1)k = h−(n+1)Lkh

−n,

from which follows h−1Lkh = Lk, and so h∈N(Lk). From this follows that q is invertible.

Let C ⊂ F0 be a chain, and set C′ = F (C). To a chain C′ ∈ H0 corresponds a sequence ofalgebraic subgroups of H which stabilizes. Therefore, C′ is finite. Now we can observe that therestriction of F to C is injective. Indeed, assume (ϕ1,M1) ≥ (ϕ2,M2), M1 ≥M2 and M2 ≥M1.Let p : M1 →M2 be the Hk-equivariant map such that ϕ2 = p ◦ ϕ1. We have observed that pmust be an isomorphism: p−1 ◦ ϕ2 = ϕ1, which shows that (ϕ2,M2) ≥ (ϕ1,M1). Finally, weconclude that C is finite, and therefore has a maximal element. By Zorn’s lemma, we concludethat F0 has a maximal element.

Now we show that for c1, c2 ∈ F0, there exists a ∈ F0 with a ≥ c1 and a ≥ c2. Let(ϕ2,M2), (ϕ1,M1) ∈ F0. Let Γi < Λ be such that ϕi is Γi-equivariant and Γi is commen-surable with Γ. Set Γ3 := Γ1 ∩ Γ2: this group is commensurable with Γ. Consider the map

ϕ3 : G/P → M1 ×M2

x 7→(ϕ1(x), ϕ2(x)

)which is Γ3-equivariant measurable. It follows from theorem 9.1 and from the ergodicity ofΓ3 on G/P that the essential image of ϕ3 is contained in one Hk-orbit, say M3, containedin M1 × M2. Using the projections on each factor, we see that (ϕ3,M3) ≥ (ϕ1,M1) and(ϕ3,M3) ≥ (ϕ2,M2) �

Let θ : G/P → N , N = Hk/Lk be a versal element of F given by theorem 10.1. Let L′ be thenormalizer in H of the Zariski closure of Lk, and q : Hk/Lk → Hk/L

′k the canonical projection.

COROLLARY.— The map θ : G/P → Hk/L′k is Λ-equivariant and measurable.

Proof.— For λ ∈Λ and (ϕ,M) ∈F , one defines via the formula λ∗ϕ(x) := π(λ)ϕ(λ−1x), anaction of Λ on F which preserves the preorder on F . Now (θ,N) being maximal, the mapsλ∗θ and θ differ by an Hk-equivariant map N → N . This implies that θ is Λ-equivariant. �

40

Proof of the Commensurator Superrigidity. Let θ : G/P →Wk, W = H/L′, be the map given

by corollary 10.2. Since π(Λ)Z

= H and θ is Λ-equivariant, we have W = Essv(θ)Z

. Defineθ : G→ F (G,Wk) by

θ(g)(x) := θ(gxP ).

One verifies that for the topology of convergence in measure on F (G,Wk), the map θ iscontinuous. It is also Λ-equivariant. Since Λ is dense in G, it acts ergodically on G, andtherefore there exists an Hk-orbit O ∈ F (G,Wk) such that θ(g) ∈O for almost every g ∈G.One deduces then from the fact that O is open in O and θ is continuous, that θ(G) ⊂ O. Inparticular, O = Hk∗θ. Since Essv(θ) is Zariski dense in W , StabHk(θ) fixes pointwise W andthus is contained in Z

(Hk

), since H is k-almost simple. We thus get a well defined continuous

map

h : G→ Hk/Z(Hk

)such that θ(g) = h(g)∗θ for all g ∈ G. One verifies that h is a homomorphism; being Λ-equivariant, it provides the desired extension. �

11. HIGHER RANK SUPERRIGIDITY

Let G be a connected R-almost simple group with real rank ≥ 2, Γ a lattice in G := G◦R,H a connected k-almost simple group and π : Γ → Hk a homomorphism with unboundedand Zariski dense image. Then there exists a proper k-subgroup L < H and a Γ-equivariantmeasurable map

ϕ : G/P → Hk/Lk

which is not essentially constant. Here, P = P(R), where P is a minimal parabolic subgroupof G. The main point of the proof consists in showing that under these conditions, one hask = R or C and ϕ is essentially rational.

Now, let P− be an R-parabolic opposite P, S < P∩P− a maximal R-split torus, t∈S := S(R),with t 6= 1. Ct is the centraliser of t ∈G, and Cu

t is N− ∩ Ct, where N− is the unipotentradical of P−.

PROPOSITION 11.1.— For almost every g∈G, the map

Cut (R) → Hk/Lkn 7→ ϕ(gnP )

is essentially continuous. If k = R or C, this map is essentially rational; otherwise, it isessentially constant.

Proof.— Set C := Ct(R). Composing the map

G× C → G/P(g, c) 7→ ϕgcP

with ϕ, and using Fubini’s theorem, we deduce that for almost every g∈G, the map

ϕg : C → Hk/Lkc 7→ ϕ(gc)

41

is measurable, and

φ : G/〈t〉 → F(C,Hk/Lk

)g 7→ ϕg

is Γ-equivariant measurable. Since Γ acts ergodically on G/〈t〉 (theorem 7.2), we deduce thatthere exists ψ∈F

(C,Hk/Lk

)such that

φ(g)∈Hk∗ψ, for almost every g∈G.

Using again Fubini’s theorem, we get that for almost every g∈G, and almost every c∈Cφ(gc), φ(g)∈Hk∗ψ,

and thus there exists hg(c)∈Hk such that

(?) φ(gc) = hg(c)∗φ(g).

Fixing g, c such that (?) holds, we get

(??) ϕg(cc′) = hg(c)ϕg(c

′), for almost every c′∈C.

Define

Vg := Essv(ϕg)Z

; Ng := {h∈H : hVg = Vg};Zg := {h∈H : h |Vg= idVg} and Hg := Ng/Zg.

Using (?) and (??), we see that hg : C → Hg(k) is a well defined homomorphism, and thuscoincides almost everywhere with a continuous one. This proves the continuity assertion.

If now k 6= R,C, then the map is constant since Cut (R) is connected. If now k = R or C, the

homomorphism hg is C∞. Furthemore, since Cut is a unipotent subgroup of the semisimple

part of Ct, hg(Cut ) < Hg is unipotent and hence hg |Cut is polynomial. In particular, the map

is essentially rational. �

To conclude rationality of ϕ on all of G/P , we need the following two lemmas.

LEMMA 11.2.— There are t1, ... tn∈S \ {e}; and connected (unipotent) R-subgroups Ui < Cutisuch that

(i) The product map

n∏i=1

Ui → N− is an R-isomorphism of varieties.

(ii)n∏i=r

Ui /n∏

i=r−1

Ui < N−.

LEMMA 11.3.— Let k = R,C, V be a k-variety and f : Rm ×Rn → V be a measurable mapsuch that for almost every x∈Rm

fx : Rn → Vy 7→ f(x, y)

is essentially k-rational, and so is

fy : Rm → Vx 7→ f(x, y)

for almost every y∈Rn. Then f is essentially rational.

Reference.— This is [Z, theorem 3.4.4]. �

42

COROLLARY 11.4.— The field k is R or C and ϕ : G/P → Hk/Lk is essentially rational.

Proof.— Take a parametrization of N− given by lemma 11.2. We prove by induction on r ≤ n,that for almost every g∈G, the map

n∏i=r

Ui(R)→ Hk/Lk

n 7→ ϕ(gnP )

is essentially rational if k = R or C, and is essentially constant otherwise. For r = n, this isthe content of proposition 11.1. Assuming it true for r, consider

Ur−1(R)×n∏i=r

Ui(R)→ Hk/Lk

(u, v) 7→ ϕ(guvP )

which by proposition 11.1 is essentially rational if k = R or C; essentially constant otherwise,

in variable v, for almost all g∈G and u∈Ur−1(R). Since Ur−1(R) normalizes

n∏i=r

Ui(R), we

get the same assertion in the variable u∈Ur−1(R). Thus, if k = R or C, lemma 11.3 impliesthe rationality of

n∏i=r−1

Ui(R)→ Hk/Lk

n 7→ ϕ(gnP )

and, if k 6= R or C, Fubini’s theorem implies that the map above is essentially constant. Thus,we get that

N−(R)→ Hk/Lkn 7→ ϕ(gnP )

is essentially constant if k 6= R and C. Since N−(R) parametrizes the big cell in G/P whichis of full measure, we get that ϕ is essentially constant. A contradiction. Thus k = R or C,and the last map is essentially rational. �

Proof of the superrigidity theorem.— We have now a rational map ϕ : G/P → H/L which isΓ-equivariant. Consider the graph

Gr(π) := {(γ, π(γ)

)∈G×H}γ∈Γ,

and Gr(π)Z

its Zariski closure in G × H. Since Γ is Zariski dense in G and π(Γ) is Zariski

dense in H, the projection of Gr(π)Z

on each factor is surjective. Now let R be the spaceR(G/P, H/L) of rational maps, endowed with the action of G×H defined by

(g, h)∗ψ(x) := hψ(g−1x), ψ∈R.

Since StabG×H(ϕ) is an algebraic subgroup, we have that Gr(π)Z

fixes ϕ. Furthemore, if

(e, h1) and (e, h2) are in Gr(π)Z

, then h1ϕ(x) = h2ϕ(x) for all x∈G/P . Since the image of ϕis Zariski dense in H/L, we get that h−1

2 h1 fixes pointwise H/L and hence

h−12 h1∈

⋂h∈H

hLh−1 = {e},

43

since H is assumed to be adjoint. Thus, Gr(π)Z

is the graph of a morphism of algebraic groupsπext : G→ H, defined over k and extending π. �

12. ARITHMETICITY

In this section, we prove the higher rank arithmeticity theorem. Thus, let Γ be a lattice inG := G(R)◦, where G is connected, adjoint, R-simple of real rank ≥ 2. We consider G as asubgroup of Aut(g), where g = Lie(G), and fix a Chevalley basis of g, so that G is definedover Q (proposition 3.7). For g∈Aut(g), let Tr(g) denote the trace of g.

We first show Tr(Γ) ⊂ Q. Indeed, any σ∈Aut(C) defines a homomorphism

σ : Γ→ G(C)

with Zariski dense image. Superrigidity implies that

– either σ(Γ) < G(C) is bounded, so that |Tr(σ(γ)

)| ≤ dim(g) for any γ∈Γ;

– or there exists θσ∈Aut(g) with θσ|Γ = σ |Γ.

In particular, we have Tr(γ) = Tr(θσ(γ)

)= Tr

(σ(γ)

)for any γ ∈ Γ. Thus, for all γ ∈ Γ,

Aut(C)(Tr(γ)

)is a bounded subset of C, which implies that Tr(Γ) is contained in Q.

Now let k = Q(Tr(Γ)

)be the field generated by {Tr(γ)}γ∈Γ, let C[G] be the space of regular

functions on G, and let ρ : G→ GL(C[G]

)be the regular representation. Let TC be the linear

span of {ρ(g)Tr}g∈G and let TΓ ⊂ kΓ be the linear span of the set of right Γ-translates of

Tr |Γ∈ kΓ. Since Γ is Zariski dense in G, the restriction map R : C[G] → CΓ is injective andinduces an isomorphism

R : TG∼−→ TΓ ⊗C.

Thus, we get a Γ-invariant k-structure on the vector space TG; the group H = ρ(Γ)Z

istherefore defined over k ⊂ R, ρ : G → H is an isomorphism defined over k, and ρ(Γ) < Hk.Thus, replacing G by H, we may assume that G is defined over k and Γ < G(k).

Next, we claim that if O denotes the ring of integers of k, Γ ∩ G(O) is of finite index in Γ:let k ↪→ kv be a non Archimedean place of k, and ιv : G(k) ↪→ G(kv) be the correspondinginjection. Let Ov be the ring of integers of kv. Since Γ < G(k) is finitely generated, the setP of places for which ιv(Γ) 6⊂ G(Ov) is finite. Moreover, by superrigidity, ιv(Γ) < G(kv) isbounded for all non Archimidean places. Thus, ιv(Γ)/

(ιv(Γ) ∩G(Ov)

)is finite, which implies

since P is finite, that Γ ∩G(O) is finite index in Γ.

Thus, replacing Γ by a subgroup of finite index, we may assume Γ < G(O). Under thecanonical isomorphism

D : G(k)∼−→ Resk/Q G(Q),

the image D(Γ) lies in Resk/Q G(Z). Furthemore, using that k is the trace field of Γ, one showseasily that D(Γ) is Zariski dense in Resk/Q G. Let Resk/Q G = L × U be the decompositioninto R-factors, such that U(R) is compact and L(R) is without compact factor. Let ∆ be theimage in L(R) of Resk/Q G(Z); ∆ is a lattice in L(R). Let q be the composition of D with

44

the projection on L(R). Observe that the projection of q(Γ) on every R-simple factor of L(R)is unbounded. Thus, q extends to an isomorphism of R-groups

qext : G∼−→ L,

such that qext(Γ) < ∆.

Thus, if p denotes the composition of the projection Resk/Q G(R) → L(R) with the inverseof qext, we get

p : Resk/Q G(R)→ G,

continuous surjective with compact kernel and such that Γ < p(Resk/Q G(Z)

). �

REFERENCES

[A′C-B] N. A′CAMPO, M. BURGER, Reseaux arithmetiques et commensurateur, d’apres G.A.Margulis, Inventiones Mathematicæ 116 (1994), 1-25.[R] M. RAGHUNATHAN, Discrete subgroups of Lie groups, Springer, 1972.[Z] R.J. ZIMMER, Ergodic theory and semisimple groups, Birkhauser, 1984.

45