Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics....

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CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment angle complexes Construction of examples Coxeter Groups and CAT(0) metrics Mike Davis Peking University June 25, 2008 http://www.math.ohio-state.edu/ mdavis/ Mike Davis Coxeter Groups and CAT(0) metrics

Transcript of Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics....

Page 1: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Coxeter Groups and CAT(0) metrics

Mike Davis

Peking University

June 25, 2008http://www.math.ohio-state.edu/ mdavis/

Mike Davis Coxeter Groups and CAT(0) metrics

Page 2: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The plan:First, explain Gromov’s notion of a nonpositively curvedmetric on a polyhedral complex. Then give a simplecombinatorial condition for nonpositive curvature on acubical cell complex.The condition is that the link of eachvertex is a “flag complex.”Next, explain a construction for producing examples ofcubical complexes where the link of each vertex is arbitrary.Give applications in topology and geometric group theory.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

1 CAT(0) geometryThe CAT(κ)-inequalityPolyhedra of piecewise constant curvature

2 Cubical complexes and simplicial complexesSimplicial complexesCubical cell complexesThe complex PL

3 A digression: moment angle complexes

4 Construction of examplesThe basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

“CAT(0)-space” is a term invented by Gromov.Also, called “Hadamard space.” Roughly, a space which is“nonpositively curved” and simply connected.

C = “Comparison” or “Cartan”A = “Aleksandrov”T = “Toponogov”

Mike Davis Coxeter Groups and CAT(0) metrics

Page 5: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

In the 1940’s and 50’s Aleksandrov introduced the notion of a“length space” and the idea of curvature bounds on lengthspace. He was primarily interested in lower curvature bounds(defined by reversing the CAT(κ) inequality). He proved that alength metric on S2 has nonnegative curvature iff it is isometricto the boundary of a convex body in E3. First Aleksandrovproved this result for nonnegatively curved piecewise Euclideanmetrics on S2, i.e., any such metric was isometric to theboundary of a convex polytope. By using approximationtechniques, he then deduced the general result (including thesmooth case) from this.One of the first papers on nonpositively curved spaces was a1948 paper of Busemann. The recent surge of interest innonpositively curved polyhedral metrics was initiated byGromov’s seminal 1987 paper.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 6: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

DefinitionsLet (X ,d) be a metric space. A path c : [a,b]→ X is ageodesic if d(c(s), c(t)) = |s − t |, ∀ s, t ∈ [a,b]. (X ,d) is ageodesic space if any two points can be connected by ageodesic segment.Given a path c : [a,b]→ X , its length, l(c), is defined by

l(c) := sup{n∑

i=1

d(c(ti−1, ti)},

where a = t0 < t1 < · · · tn = b runs over all possiblesubdivisions.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 7: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

For κ ∈ R, X2κ is the simply connected, complete, Riemannian

2-manifold of constant curvature κ:X2

0 is the Euclidean plane E2.If κ > 0, then X2

κ = S2 with its metric rescaled so that itscurvature is κ (i.e., it is the sphere of radius 1/

√κ).

If κ < 0, then X2κ = H2, the hyperbolic plane, with its metric

rescaled.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 8: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

A triangle T in a metric space X is a configuration of threegeodesic segments (the “edges”) connecting three points (the“vertices”) in pairs. A comparison triangle for T is a triangle T ∗

in X2κ with the same edge lengths. When κ ≤ 0, a comparison

triangle always exists. When κ > 0, a comparison triangleexists ⇐⇒ l(T ) ≤ 2π/

√κ, where l(T ) denotes the sum of the

lengths of the edges. (The number 2π/√κ is the length of the

equator in a 2-sphere of curvature κ.)

If T ∗ is a comparison triangle for T , then for each edge of Tthere is a well-defined isometry, x 7→ x∗, which takes the givenedge of T onto the corresponding edge of T ∗.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 9: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

A metric space X is a CAT(κ)-space if:

If κ ≤ 0, then X is a geodesic space, while if κ > 0, it isrequired there be a geodesic segment between any twopoints < π/

√κ apart.

(The CAT(κ) inequality). For any triangle T (withl(T ) < 2π/

√κ if κ > 0) and any two points x , y ∈ T , we

haved(x , y) ≤ d∗(x∗, y∗),

where x∗, y∗ are the corresponding points in thecomparison triangle T ∗ and d∗ is distance in X2

κ.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 10: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

T T

y

x

y

x

*

*

*

d(x , y) ≤ d∗(x∗, y∗),

Mike Davis Coxeter Groups and CAT(0) metrics

Page 11: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

DefinitionA metric space X has curvature ≤ κ if the CAT(κ) inequalityholds locally.

FactsIf κ′ < κ, then CAT(κ′) =⇒ CAT(κ).CAT(0) =⇒ contractible.curvature ≤ 0 =⇒ aspherical.

Theorem (Aleksandrov and Toponogov)

A Riemannian mfld has sectional curvature ≤ κ iff CAT(κ)holds locally.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

A convex polytope in Xnκ is an Xκ-polytope.

DefinitionSuppose F is the poset of faces of a cell complex. An Xκ-cellstructure on F is a family (CF )F∈F of Xκ-polytopes s.t.whenever F ′ < F , CF ′ is isometric to the corresponding face ofCF .

Mike Davis Coxeter Groups and CAT(0) metrics

Page 13: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

Suppose Λ is a Xκ-polyhedral complex. A path c : [a,b]→ Λ ispiecewise geodesic if there is a subdivisiona = t0 < t1 · · · < tk = b s.t. for 1 ≤ i ≤ k , c([ti−1, ti ]) iscontained in a single (closed) cell of Λ and s.t. c|[ti−1,ti ] is ageodesic. The length of the piecewise geodesic c is defined byl(c) :=

∑ni=1 d(c(ti), c(ti−1)).

Λ has a natural length metric:d(x , y) := inf{l(c) | c is a piecewise geodesic from x to y }.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

DefinitionThe geodesic space Λ is a piecewise constant curvaturepolyhedron. As κ = +1,0,−1, it is, respectively, piecewisespherical, piecewise Euclidean or piecewise hyperbolic.

Piecewise spherical (= PS) polyhedra play a distinguished rolein this theory. In any Xκ-polyhedral cx each “link” naturally hasa PS structure.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

The link of a vertex v in a cell is the intersection of a smallsphere about the vertex with the cell.For example, the link of a vertex in a cube is a (spherical)simplex.

Lk(v)Lk(v)

Similarly, the link, Lk(v), of a vertex in a cubical cell complex isa simpicial cx.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 16: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The CAT(κ)-inequalityPolyhedra of piecewise constant curvature

TheoremLet Λ be an Xκ-polyhedral complex. TFAE

curv(Λ) ≤ κ.∀v ∈ Vert Λ), Lk(v ,Λ) is CAT(1).

Mike Davis Coxeter Groups and CAT(0) metrics

Page 17: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

DefinitionAn affine simplex is the convex hull of a finite set T of affinelyindependent points in some Euclidean space. Its dimension isCard(T )− 1. For S a finite set, ∆S, the simplex on S, is theconvex hull of the standard basis of RS (whereRS := {x : S → R}).

ExampleA 1-simplex is an interval; a 2-simplex is a triangle; a 3-simplexis a tetrahedron.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

DefinitionAn abstract simplicial complex consists of a set S (of vertices)and a poset S of finite subsets of S s.t.

∅ ∈ S,{s} ∈ S, ∀s ∈ S.If T ∈ S and U ⊂ T , then U ∈ S.

DefinitionA (geometric) simplicial complex is a cell complex in which allcells are geometric simplices.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

Suppose L is a geometric simplicial cx. Put

S := Vert(L), andS(L) := {T ⊂ S | T is the vertex set of a simplex in L}.

DefinitionA geometric realization of an abstract simplicial cx S is ageometric simplicial cx L s.t. S = S(L).

TheoremEvery abstract simplicial cx S has a geometric realization.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

Given a set S and a function x : S → R,Supp(x) := {s ∈ S | xs 6= 0}. RS denotes the Euclidean spaceof finitely supported functions x : S → R and ∆S is the simplexon S. We want to prove:

TheoremEvery abstract simplicial cx S has a geometric realization.

Proof.

Given S, define a subcx L ⊂ ∆S by

L := {x ∈ ∆S | Supp(x) ∈ S} =⋃

T∈S∆T

Clearly, S(L) = S.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

All right simplicial complexes

DefinitionA spherical simplex σ is all right if it is isometric to theintersection of spheren with the “quadrant”, [0,∞)n+1.Equivalently, σ is all right if each of its edge lengths is π/2 (or ifeach of its dihedral angles is π/2.

A simplicial complex L is all right if each of its simplices is allright. Thus, every simplicial complex has an all right PSstructure (defined by declaring each of its simplices to be allright).

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

Definition

The standard cube on a set S is: �S := [−1,1]S ⊂ RS. Itsdimension is Card(S).

For each T ⊂ S, put

�T := [−1,1]T × {0}S−T ⊂ RS.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

The link of a vertex in a cube is a (spherical) simplex. Similarly,the link, Lk(v), of a vertex in a cubical cell complex is asimpicial cx.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

Suppose v is a vertex in a cubical cell complex P.As an abstract simplicial complex Lk(v ; P) is isomorphic tothe poset of cells of P which contain v .A neighborhood of v in P is homeomorphic to the cone onLk(v).

Mike Davis Coxeter Groups and CAT(0) metrics

Page 25: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

DefinitionA simplicial cx L is a flag complex iff any finite set of verticeswhich are pairwise connected by edges spans a simplex of L.

Examples

∂∆n is not a flag cx for n ≥ 2A k -gon (i.e. a triangulation of S1) is a flag cx iff k ≥ 4The barycentric subdivision of any cell complex is a flag cx.(This shows that the condition of being a flag cx does notrestrict the topological type of L: it can be any polyhedron.)

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

Gromov’s LemmaA simplicial complex with its all right PS structure is CAT(1)⇐⇒ it is a flag complex.

CorollaryA cubical complex with its natural piecewise Euclidean metric isnonpositively curved ⇐⇒ the link of each vertex is a flagcomplex.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

Recall �T := [−1,1]T × {0}S−T ⊂ RS.A face of �S is parallel to �T if it has the form [−1,1]T × {ε} forsome ε ∈ {±1}S−T .

The cubical complex PL

Given a simplicial complex L with vertex set S, define asubcomplex PL of �S by

PL :=⋃

T∈S(L)

all faces parallel to �T

Main Property

For each vertex v , Lk(v ,PL) = L

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

ExampleSuppose S consists of 3 points and L isthe union of an interval and a point. Then�S is a 3-cube and PL is indicated subcx.

LPL

More examples

If L = ∆n−1, then PL = �n

If L = ∂∆n−1, then PL = ∂�n = Sn−1.If L is a set of n points, then PL is the 1-skeleton of �n,eg, if L = S0, then PL = ∂�2 = S1.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

Joins

If T and U are disjoint sets, then ∆T ∗∆U = ∆T∪U . (Note:dim(∆T ∗∆U) = Card(T ∪ U)− 1 = dim(∆T ) + dim(∆U) + 1.)Similarly, If L1 and L2 are simplicial complexes, then L1 ∗ L2 isdefined by taking the joins of simplices in L1 with those in L2(including the two empty simplices). We have:S(L1 ∗ L2) = S(L1)× S(L2)

More

P(L1∗L2) = PL1 × PL2 , eg, if L = S0 ∗ S0, thenPL = S1 × S1 = T 2, or if L is the n-fold join S0 ∗ · · · ∗ S0

(the bdry of an n-dim octahedron), then PL = T n.If L is a k -gon, then PL is the orientable surface of Eulercharacteristic 2k−2(4− k)

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

If L is a triangulation of Sn−1, then PL is an n-mfld.

The (Z/2)S-action

Let {rs}s∈S be standard basis for (Z/2)S. Represent rs asreflection on �S across the hyperplane xs = 0, ie, rschanges the sign of the sth-coordinate. This defines a(Z/2)S-action on �S.The subcomplex PL is (Z/2)S-stable.[0,1]S is a (strict) fundamental domain for (Z/2)S-action on�S (= [−1,1]S).K := PL ∩ [0,1]S is a (strict) fundamental domain for(Z/2)S-action on PL.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

The universal cover of PL is denoted PL. The cubical structureon PL lifts to one on PL.

LPL

PL˜

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

Let WL be the gp of all lifts of elements of (Z/2)S to PL.Let ϕ : WL → (Z/2)S be the projection. We have the shortexact sequence

1→ π1(PL)→WL → (Z/2)S → 1.(Z/2)S acts simply transitively on Vert(PL), so, WL actssimply transitively on Vert(PL).

Let v ∈ K be the vertex (1, . . . ,1). Choose a lift K of K inPL (N.B. K is a cone) and let v be the lift of v in K . The1-cells at v or v correspond to elements of S. Thereflection rs flips the 1-cell at v labeled by s. Let s be theunique lift of rs which stabilizes the corresponding 1-cell atv . (Eventually, I will drop the ˜ from s.)

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

A presentation for WL

Since s2 fixes v and covers the identity on PL, it followsthat s2 = 1.Since WL is simply transitive on Vert(PL), the 1-skeleton ofPL is the Cayley graph of (WL, S).Suppose {s, t} is an edge of L. The corresponding 2-cell atv has edges labeled successively by s, t , s, t . It follows that(st)2 = 1.

Since the 2-skeleton of PL is simply connected, it is theCayley 2-complex of a presentation. Therefore, WL has apresentation with generating set S = {s}s∈S and relations:s2 = 1 and (st)2 = 1, ∀{s, t} ∈ Edge(L).WL is a right-angled Coxeter group.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 34: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Simplicial complexesCubical cell complexesThe complex PL

Theorem

PL is contractible iff L is a flag cx.

Proof.Gromov: a cubical complex is CAT(0) ⇐⇒ it is simplyconnected and the link of each vertex is a flag cx. Since the2-skeleton of PL is the Cayley two complex for (WL,S), PL issimply connected.

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Here is a generalization of the construction of PL which hasreceived a good deal of recent interest.

Let (X ,A) be a pair of spaces and L a simplicial cx withVert(L) = S. We define certain subspaces of the product∏

s∈S X .For each T ∈ S(L)>∅, let X T be the set of (xs)s∈S in theproduct defined by {

xs ∈ X if s ∈ T ,xs ∈ A if s /∈ T .

Z (L; X ,A) :=⋃

T∈S(L)>∅

X T

Mike Davis Coxeter Groups and CAT(0) metrics

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CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

Examples

(X ,A) = ([−1,1], {±1}). Then Z (L; [−1,1], {±1}) = PL.(X ,A) = (S1, {1}). Then the fundamental gp ofZ (L; S1, {1}) is the right-angled Artin gp determined by the1-skeleton of L. If L is a flag cx, then Z (L; S1, {1}) is thestandard K (π,1) for the Artin gp.(X ,A) = (D2,S1). Then Z (L; D2,S1) is the moment anglecx of L. The group (S1)S acts on (Z (L; D2,S1). Thequotient space is the same space K (⊂ [0,1]S) consideredearlier. If K is a n-dim convex polytope and L is the bdrycx of its dual, then Z (L; D2,S1) is a smooth mfld, and if Tis an appropriate subgp of codim n in (S1)S, thenZ (L; D2,S1)/T is a “toric variety”.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 37: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

L is a flag cx with vertex set S and WL is associatedright-angled Coxeter gp. S is its fundamental set of generators..

The basic constructionA mirror structure on a space X is a family of closed subspaces{Xs}s∈S. For x ∈ X , put S(x) = {s ∈ S | x ∈ Xs}. Define

U(W ,X ) := (W × X )/ ∼,

where ∼ is the equivalence relation: (x ,w) ∼ (x ′,w ′) ⇐⇒x = x ′ and w−1w ′ ∈WS(x) (the subgp generated by S(x)).U(W ,X ) is formed by gluing together copies of X (thechambers). The gp WL (= W ) acts on it.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 38: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

Another construction of PL

Recall K := PL ∩ [0,1]S. For each s ∈ S, Ks is the intersectionof K with the hyperplane xs = 0. This is a mirror structure on K .

Theorem

The natural maps U((Z/2)S,K )→ PL and U(WL,K )→ PL arehomeomorphisms.

The basic ideaThe topology of the simplicial cx L is reflected in properties ofthe gp WL.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 39: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

Suppose π is a torsion-free gp. Its cohomologicaldimension, cd(π) is defined to be the maximum integer k stHk (π; M) 6= 0 for some π-module M.Its geometric dimension, gd(π) is the smallest dimensionof a K (π,1) complex. Obviously, cd(π) ≤ gd(π).Eilenberg-Ganea proved equality if cd(π) ≥ 3 andStallings, Swan proved it for cd(π) = 1.

The Eilenberg-Ganea Problem

Is there a gp π with cd(π) = 2 and gd(π) = 3?

Conjectured answerYes.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 40: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

Suppose L is a flag triangulation of an acyclic 2-complexwith π1(L) 6= 0. Put πL := π1(PL) = Ker(ϕ : WL → (Z/2)S.πL is torsion-free. It is a conjectured Eilenberg-Ganeacounterexample.

Put ∂K := K −◦K . In our case it is acyclic. It follows that

U(WL, ∂K ) is acyclic (but not simply connected). Hence,cd(πL) = 2.

dim PL = dim L + 1 = 3 and the only contractible complexwhich πL seems to act on is PL (= U(WL,K )).

RemarkBrady, Leary, Nucinkis proved these WL are counterexamplesto the version of the Eilenberg-Ganea Problem for groups withtorsion.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 41: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

Example (Different cd over Z than Q)

Suppose L is a flag triangulation of RP2. ThenH3(πL; ZπL) = H3(PL; Z) = H2(RP2) = Z/2.

H3(PL; Q) = 0 and H2(PL; Q) is a countably generated Qvector space. Hence,cdZ(πL) = 3 and cdQ(πL) = 2

Mike Davis Coxeter Groups and CAT(0) metrics

Page 42: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

FactsA closed m-mfld Mm, m ≥ 3, with the same homology asSm need not be homeomorphic to Sm, because it need notbe simply connected. However,If π1(Mm) = 1, then Mm ∼= Sm (Poincare Conjecture).Similarly, a contractible open mfld Y m, m ≥ 3, ishomeomorphic to Rm iff it is simply connected at∞.(Stallings, Freedman Perelman).Every such homology m-sphere Mm (simply connected ornot) bounds a contractible (m + 1)-mfld.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 43: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

Suppose Ln−1 is a non simply connected homology(n − 1)-sphere triangulated as a flag cx.

Then PL is a contractible n-dim homology mfld (the nonmanifold points are the vertices) and PL is not simplyconnected at∞. (Its fundamental gp at∞ is the inverselimit of free products of an increasing number of copies ofπ1(L).)

PL can be modified to be a contractible n-mfld. Let C be a

contractible n-mfld bounded by L (= ∂K ). Remove◦K and

replace it by◦C. Then Y n := U(WL,C) is a contractible

n-mfld 6∼= Rn and Mn := Y n/π (where π = π1(PL)) is aclosed aspherical mfld with universal cover Y n.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 44: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

Reflection Group Trick

Given a group π which has a finite K (π,1) complex, this isa technique for constructing an aspherical mfld M whichretracts back onto K (π,1). (Aspherical means its universalcover is contractible.) In a nutshell the trick goes as follows:Thicken K (π,1) to a X , a compact mfld with bdry. (X ishomotopy equivalent to K (π,1).)Put L := ∂X . Triangulate L as a flag cx and let W (= WL)be the corresponding right-angled Coxeter gp. As before

modify PL to a mfld by removing each copy of◦K and

replacing it by◦X

M := U((Z/2)S,X ) is the desired aspherical mfld.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 45: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

Sample applications

Point: many interesting gps have finite K (π,1)-complexes (even2-dimensional ones).

By choosing π a Baumslag-Solitar gp, we can get π1(M)

- to be non-residually finite, or- to have an infinitely divisible subgp (∼= Z[1/2]).

By choosing π to have unsolvable word problem (can dothis with a 2-dim K (π,1)), we get π1(M) with unsolvableword problem.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 46: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

M := U((Z/2)S,X ).As before, we can construct U(WL,X ). It is notcontractible. But it is aspherical. Hence, M is aspherical(since it is covered by U(WL,X )). (Pf: It is a union ofcopies of X glued together along contractible pieces.)M = (univ cover of M). We can explicitly describe M asfollows. Let X = (univ cover of X ) and π = π1(X ).

L is the induced triangulation of ∂X and S = Vert(L). W(= WeL) the corresponding right-angled Coxeter gp. Give Xthe induced mirror structure (indexed by S). ThenM = U(W , X ).

Mike Davis Coxeter Groups and CAT(0) metrics

Page 47: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

The gp W o π acts on U(W , X ) with quotient space X and if Γ

is the inverse image of the commutator subgp of WL in W , thenΓ o π acts freely with quotient space M.

Mike Davis Coxeter Groups and CAT(0) metrics

Page 48: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

Mike Davis Coxeter Groups and CAT(0) metrics

Page 49: Coxeter Groups and CAT(0) metrics · 2008. 6. 24. · Mike Davis Coxeter Groups and CAT(0) metrics. CAT(0) geometry Cubical complexes and simplicial complexes A digression: moment

CAT(0) geometryCubical complexes and simplicial complexes

A digression: moment angle complexesConstruction of examples

The basic constructionCohomological dimensionAspherical mflds not covered by Rn

The reflection group trick

References

M.W. Davis, Exotic aspherical manifolds, School onHigh-Dimensional Topology, ICTP, Trieste, 2002.

M.W. Davis, The Geometry and Topology of CoxeterGroups, London Math. Soc. Monograph Series, vol. 32,Princeton Univ. Press, 2007.

M.W. Davis The cohomology of a Coxeter group with groupring coefficents, Duke Math. J. 91 (1998), 297–314.

Mike Davis Coxeter Groups and CAT(0) metrics