On the virtual cohomological dimensions of Coxeter groups

A.N.Dranishnikov

Abstract. We apply Bestvina’s approach of calculation of the virtual cohomological dimension of Coxeter groups. The explicit formula for vcd_{F} Γ in terms of cohomological properties of corresponding complex K is given.

Math. Subj. Class. Primary: 20F32, 55M10, 57S30 Secondary: 16A60, 20J05, 51K10, 54E45

1. Introduction

If an Eilenberg MacLane complex K(Γ, 1) of a group Γ is finite then the cohomological dimension cd_{F} Γ of the group Γ with respect to an abelian group F can be defined as a maximal number n such that H_{c}^{n}(X̄; F) ≠ 0, where X̄ is the universal cover of K(Γ, 1). According to the theorem of Serre if cd_{F} Γ < ∞ for a subgroup Γ of the finite index of a group G then the number cd_{F} Γ does not depend on a choice of Γ. That number is called the virtual cohomological dimension of G with respect to F and is denoted by vcd_{F} Γ.

For every Coxeter group Γ Bestvina has constructed a finite polyhedron B(Γ, F) such that vcd_{F} Γ = dimB(Γ, F) for sufficiently good F [1]. Using his result we give a formula for vcd_{F} Γ for Coxeter groups in terms of cohomologies of Davis’ panel complex. Then we apply that formula to studying the topology of the boundaries of Coxeter groups. First we get a new proof of the fact that every Pontryagin surface can be a boundary of a Coxeter group [2] (see also [1], Remark 3). Second, the formula implies that the Boltyanski compactum cannot be a boundary. Thus, the logarithmic law can fail for virtual dimension of the product of two Coxeter groups i.e. there are groups Γ_{1} and Γ_{2} with vcd(Γ_{1} × Γ_{2}) < vcdΓ_{1} + vcdΓ_{2}. Nevertheless the logarithmic law holds if Γ_{1} = Γ_{2}. So this behavior of vcd is analogous to that of the covering dimension dim for ANR-compacta [3]. The third application of the formula is an attempt to construct a Coxeter group Γ with 3-dimensional boundary ∂Γ having dim_{ℚ}∂Γ = 1. Note that groups with 2-dimensional and rationally 1-dimensional boundaries were constructed before [4], [1], [2].

The following problem is still unsolved for n ≥ 3.

Problem. Does there exist a Coxeter group Γ_{n} with n-dimensional boundary and with dim_{ℚ}∂Γ_{n} = 1?

That problem for compacta without any relation to groups was solved for n = 2 by Pontryagin in early 30s, the corresponding compactum X_{n} for n = 3 was constructed by Kuzminov [5] only in 60s, and in 80s it was done for all n [3].

Let V be a finite set and let I ⊂ V × V be a symmetric subset. The group Γ given by the following presentation

⟨V | v^{2} = 1, (vw)^{m(v,u)} = 1 ∀v ∈ V ∀(v, w) ∈ I; 0 < m(v, w) = m(w, v) ∈ ℤ⟩

is called the Coxeter group. If m(v, w) = 2 for all (v, w) ∈ I, then the groupis called the right-angled Coxeter group. The group Γ together with its presentation is called Coxeter system (Γ, V). Let F be the set of all non-empty subsets of V that generate a finite subgroup of Γ. Then F is a partially ordered set (poset) by inclusion. Then [6] F is isomorphic to the poset of simplices of some simplicial complex K = K(Γ, V). In the case of right-angled Coxeter group we can construct K(Γ, V) by the following rule. The vertices of K are V. Two vertices v and w are joined by an edge if (v, w) ∈ I. A set of n + 1 vertices spans an n-dimensional simplex iff all vertices are pair wise joined by edges [6]. The last property of a simplicial complex gives a definition of a flag complex [6].

2. The local cohomological dimension of simplicial complex and the formula for vcd_{F} Γ

Let K be a simplicial complex. We recall that the collection of all simplices in K containing a given vertex v forms a complex St(v, K) called the star of v in K and the collection of all simplices in St(v, K) do not containing v forms a complex Lk(v, K), called the link of v in K. We denote by β^{1}K the first barycentric subdivision of K. For every simplex σ ⊂ K we define the normal star nst(σ, K) of σ in K as a subcomplex of the star St(v(σ), β^{1}K) of the barycenter v(σ) of the simplex σ in β^{1}K consisting of those simplices δ whose intersection with β^{1}σ consists of v(σ). The normal link nlk(σ, K) of σ in K is the boundary of nst(σ, K) i.e. the subcomplex of nst(σ, K) consisting of all simplices having empty intersection with v(σ).

We are going to use the same symbol for a simplicial complex and for its geometric realization.

Definition. The local cohomological dimension of a simplicial complex K with respect to a coefficient group G is

lcd_{G}K = max_{σ∈K}{m | H^{m}(nst(σ, K), nlk(σ, K); G) ≠ 0}.

The global cohomological dimension [7] of a simplicial complex K over G is cd_{G}K = max{m | H^{m}(K; G) ≠ 0}. Note that H^{m}((nst(σ, K), nlk(σ, K)); G) = H^{m−1}(nlk(σ, K); G) and hence lcd_{G}K = max_{σ∈K}{m | H^{m−1}(nlk(σ, K); G) ≠ 0}.

These dimensions can differ from each other. For example, if K is a triangulation of the join ℝP^{2} ∗ ℝP^{2} of the projective plane with itself, generated by some triangulation on ℝP^{2}, then its dimensions over the integers and the rationals are cd_{ℤ}K = lcd_{ℤ}K = 5 and lcd_{ℚ}K = 4 and cd_{ℚ} = 0.

Theorem 1. For every abelian group G and every finite-dimensional simplicial complex K there is the inequality lcd_{G}K ≥ cd_{G}K.

We denote by CK the cone over a simplicial complex K with the natural triangulation.

Theorem 2. For every Coxeter group Γ defined by a Coxeter system (Γ, V) there is the equality vcd_{F} Γ = lcd_{F} CK where K = K(Γ, V) and F is an additive group of field or the group of integers.

Every r-dimensional complex K has a filtration L_{0} ⊂ L_{1} ⊂ ... ⊂ L_{r} = K where L_{i} = ⋃_{dimσ=r−i} nst(σ, K). By induction we define the filtration L′_{0} ⊂ L′_{1} ⊂ ... ⊂ L′_{r} = B(K, F) for a given abelian group F. We define polyhedra L′_{i} together with projections p_{i} : L′_{i} → L_{i} such that p_{i} is an extension of p_{i−1} for every i. Define L′_{0} = L and p_{0} = id. Assume that p_{i−1} : L′_{i−1} → L_{i−1} is defined. For every (r − i)-simplex σ we consider A(σ) = p_{i−1}^{−1}(nlk(σ, K)). Let C(σ) be an F -acyclic polyhedron containing A(σ) and of the least possible dimension. We attach C(σ) to L′_{i−1} along A(σ) and extend the map p_{i−1} over C(σ) such that the image of C(σ) − A(σ) lies in nst(σ, K) − nlk(σ, K). Do that for all σ to obtain p_{i} : L′_{i} → L_{i}. The resulting space L_{r} is Bestvina’s complex B(K, F) of a complex K with respect to a group F.

Note that the map p = p_{r} : B(K, F) → K has the property that the preimage of every normal star nst(σ, K) is F -acyclic.

Theorem3 [1]. For every Coxeter group Γ defined by a Coxeter system (Γ, V) the following equality holds vcd_{F} Γ = dimB(CK(Γ, V); F) if F is a field or the group of integers.

Proposition 1. For every simplex σ ⊂ K, H^{n}(nlk(σ, K); F) = H^{n}(p^{−1}(nlk(σ, K); F) for all n.

Proof: First, we define by induction the notion of a star-subcomplex N of K. The empty set is a star-subcomplex. A union of some normal stars of (r − i)-dimensional simplices nst(σ, K) form a star-subcomplex of K if all pair wise intersections of that normal stars are star-subcomplexes. Induction, the Mayer-Vietoris sequence and the fact that H̄^{∗}(p^{−1}(nst(σ, K); F) = 0 imply H^{n}(N; F) = H^{n}(p^{−1}(N); F) for all n and for every starsubcomplex N ⊂ K. This argument is well-known and sometimes is called the combinatorial Vietoris-Begle theorem.

Proposition 2. dimB(K, F) = lcd_{F} K

Proof: Let lcd_{F} K = n. Hence H^{n−1}(nlk(σ, K); F) ≠ 0 for some simplex σ in K. By Proposition 1 we have H^{n−1}(p^{−1}(nlk(σ, K)); F) ≠ 0. Therefore, H^{n−1}(A(σ); F) ≠ 0. Then by the construction C(σ) is at least n-dimensional. Hence dimB(K, F) ≥ n.

Now let dimB(K, F) = n. then by the construction of B(K, F) there exist a simplex σ ⊂ K with H^{n−1}(A(σ); F) ≠ 0. Then by Proposition 1, H^{n−1}(nlk(σ, K); F) ≠ 0 and hence lcd_{F} K ≥ n.

Now the proof of Theorem 2 follows by Proposition 2 and Theorem 3.

Proof of Theorem 1: Assume that cd_{F} K = n. Consider Bestvina’s complex p : B(K, F) → K. Proposition 1 implies that H^{n}(B(K, F); F) ≠ 0 and, hence, dimB(K, F) ≥ n. Proposition 2 implies the proof.

Proposition 3. Let CK be a cone over K then lcd_{F} CK = max{lcd_{F} K, cd_{F} K + 1}.

Proof: For every simplex σ ⊂ K ⊂ CK the normal link nlk(σ, CK) is homeomorphic to the cone over the normal link nlk(σ, K) and, hence, is cohomologicaly trivial. The normal link of the cone vertex v is K. The normal link of any other simplex σ containing v is homeomorphic to nlk(δ, K) for some simplex δ. If cd_{F} K < lcd_{F} K then lcd_{F} CK = lcd_{F} K by the above and hence the formula is true. If i = cd_{F} K ≥ lcd_{F} K then lcd_{F} CK = i + 1 and the formula is true again.

3. Applications of the formula for vcd_{F} Γ

Theorem 4. An every Coxeter group Γ has the following properties:

a) vcd_{ℚ}Γ ≤ vcd_{G}Γ for any group G,

b) vcd_{ℤ_{p}}Γ = vcd_{ℚ}Γ for almost all primes p,

c) there exists prime p such that vcd_{ℤ_{p}}Γ = vcdΓ

Proof: a) Let vcd_{ℚ}Γ = n. By Theorem 2 there exists σ ⊂ CK such that

H^{n−1}(nlk(σ, CK); ℚ) ≠ 0. The Universal Coefficient Formula implies that the group H^{n−1}(nlk(σ, CK); ℤ) has a nontrivial rank. Hence, H^{n−1}(nlk(σ, CK); G) ≠ 0 and therefore, lcd_{G}CK ≥ n. Theorem 2 implies that vcd_{G}Γ ≥ n.

b) Let again vcd_{ℚ}Γ = n. Since we have only finitely many simplices δ ⊂ CK and finitely many m ≤ dimCK, only finitely many prime numbers are involved in the torsions of H^{m}(nlk(δ, CK); ℤ). Hence for almost all prime p, H^{m}(nlk(δ, CK); ℤ_{p}) = 0 for m ≥ n. Therefore, lcd_{ℤ_{p}}CK ≤ n. That together with a) proves b).

c) Let vcdΓ = r, then by Theorem 2 there is a simplex σ′ ⊂ CK such that

H^{r−1}(nlk(σ′, CK); ℤ)) ≠ 0. Hence by virtue of the Universal Coefficient Formula, H^{r−1}(nlk(σ′, CK); ℤ_{p}) ≠ 0. Hence, lcd_{ℤ_{p}}CK ≥ r and hence, lcd_{ℤ_{p}}CK = r. The Theorem 2 implies that vcd_{ℤ_{p}}Γ = vcdΓ.

Corollary 1. The equality vcd(Γ × Γ) = 2vcdΓ holds for every Coxeter group.

Corollary 2. A Boltyanski’s compactum can not be the boundary of a Coxeter group.

Proofs: Since the formula vcd_{F} Γ×Γ′ = vcd_{F} Γ+vcd_{F} Γ′ holds for every field F, Theorem 6 b) implies the proof of Corollary 1. The characterizing property of Boltyanski’s compactum B is dimB × B = 3 [5]. Then in view of the following theorem Corollary 2 follows.

Theorem 5 (Bestvina-Mess) [4]. Let ∂Γ be the boundary of a Coxeter group Γ then for every abelian group G, dim_{G}∂Γ = vcd_{G}Γ − 1.

We recall that a compactum X has the cohomological dimension dim_{G}X ≤ n if for every closed subset A ⊂ X the relative Cech cohomology group Ȟ^{n+1}(X, A; G) is trivial [3],[5].

Corollary 3 [2]. Every Pontryagin surface Π_{p} is the boundary of some right-angled Coxeter group.

Proof: Pontryagin surface Π_{p} is defined by the properties: dim_{ℚ}Π_{p} = dim_{ℤ_{q}}Π_{p} = 1 for q relatively prime with p and dimΠ_{p} = 2. Hence dim(Π_{p} × Π_{q}) = 3 if p and q are relatively prime. If we take the Moore complex M(ℤ_{p}, 1) with an arbitrary triangulation K then by Theorem 2 vcd_{F} Γ = lcd_{F} CK for the Coxeter group Γ generated by K. By Proposition 3 lcd_{F} CK = max{lcd_{F} K, cd_{F} K + 1}. It is easy to see that lcd_{F} K = 2 for all F. Note that cd_{F} K = 2 if F = ℤ_{p} and cd_{F} K = 1 for F = ℚ, ℤ_{q}. Hence vcd_{F} Γ = 2 if F = ℤ_{q} or ℚ and vcdΓ = 3. Theorem 7 implies that the boundary ∂Γ is a Pontryagin surface Π_{p}.

We recall that the boundary of the Coxeter system (Γ, K) is the visual sphere at infinity of CAT(0) cubical complex Σ associated with (Γ, K) [6].

4. Complexes with a given link

In this paragraph we prove the following

Lemma 1. For every 2-dimensional compact polyhedron K there exist a finite simplicial complex L such that the link Lk(v, L) of every vertex is homeomorphic to K.

In view of Theorem 2 and Proposition 3 Lemma 1, applied to K equal to the real projective plane, almost gives a solution of the Problem for n = 3 (see the Introduction). It gives a flag complex L with cdL = 3 and lcd_{ℚ}L = 2. If additionally we could require the equality cd_{ℚ}L = 1, then the Coxeter group Γ defined by the simplicial complex L should have dimensions vcdΓ = 4 and vcd_{ℚ}Γ = 2. Hence the boundary ∂Γ would be 3-dimensional with dim_{ℚ}∂Γ = 1. Thus, to get a solution of the Problem one needs a complex L of Lemma 1 with trivial rational 2-dimensional cohomology groups.

By the definition a cubical complex Σ is the union of cubes C with intersections along faces. A locally finite cubical complex poses the following natural metric on it. All cubes are considered isomorphic to the standard euclidean cube of length one and the distance between two points in different cubes is defined as the infimum of the length of curves joining these two points.

The following theorem can be extracted from [6].

Theorem 6. For every right-angled Coxeter system (Γ, S) there exists a cubical complex Σ = Σ(Γ, S) with the following properties:

1) Γ acts by isometries on Σ with finite isotropy groups and with the orbit space homeomorphic to the cone CK over K = K(Γ, S).

2) There is the natural imbedding j of CK in Σ such that the projection onto orbit space is a retraction and hence Γ(D_{0}) = Σ, where D_{0} = j(CK).

3) Let v_{0} be the cone point of D_{0}, then D_{0} is the image of 1/2-homothety centered at v_{0} of the cubical star St(v_{0}, Σ)= the union of all cubes in Σ having v_{0} as a vertex.

4) There is a bijection s ↔ {v_{0}, s(v_{0})} between S and the set of edges of Σ emanating from v_{0} such that s as an element of Γ takes v_{0} to s(v_{0}). Edges [v_{0}, s_{1}(v_{0})], ..., [v_{0}, s_{k}(v_{0})] form a k-cube in the complex Σ if and only if (s_{i}, s_{j}) ∈ I for all i, j ≤ k, i ≠ j.

5) Γ acts freely and transitively on the set of all vertices of Σ.

6) Σ is a CAT(0) geodesic space and, hence, is contractible.

The following theorem is due to Tits and Selberg.

Theorem 7 [6]. Every Coxeter group Γ has a torsion free subgroup Γ′ of finite index.

Note that in the case of right-angled Coxeter groups the commutator group [Γ, Γ] can be taken as Γ′.

Proof of Lemma 1: Consider a flag triangulation on K with the set of vertices S and consider the corresponding Coxeter system (Γ, S). Show that the cubical complex Σ = Σ(Γ, S) of Theorem 6 admits a triangulation with the same set of vertices. In the domain 2D_{0} = Lk(v_{0}, Σ) for every 2-dimensional cube F containing v_{0} we consider a triangulation of F generated by the diagonal [v_{0}, w]. For every 3-dimensional cube C in 2D_{0} consider a triangulation generated by diagonals [v_{0}, w] in 2-faces F of C (v_{0} ∈ F) and antiparallel diagonals in opposite 2-faces F̄ (F ∩ {v_{0}} = ∅). If C and C′ are two cubes in 2D_{0} with common face F, then v_{0} ∈ F and hence the triangulations agree on these two cubes. Thus we have a triangulation on 2D_{0} with the same set of vertices as the cubical complex 2D_{0}.

Denote by G a subgroup of Γ consisting of all elements having an even length in the alphabet S. Note that G is well-defined. Indeed, since all the relations in the presentation of the group Γ have an even length, the equality s_{1}s_{2}...s_{n} = 1 is impossible for an odd n by Tits theorem [8]. Note that G(2D_{0}) = Σ. Indeed, for every g ∈ Γ and for every cube C ⊂ 2D_{0}, we have g(1/2C) ⊂ ∪_{s∈S} gs(C) ⊂ G(2D_{0}). By Theorem 6, 2) we have Σ = Γ(D_{0}) = Γ(∪1/2C) ⊂ G(2D_{0}). We define a triangulation on Σ by translations of that on 2D_{0} by the group G. To show that this definition is correct it suffices to show that for every 2-dimensional cube Φ in Σ and for any two 2-cubes F and F′ in 2D_{0} with g(F) = Φ = g′(F′), g, g′ ∈ G the images of the diagonals participating in the triangulation on 2D_{0} are the same. Consider g^{−1}g′(F′) = F. Without loss of generality we may assume that v_{0} ∈ F ∩ F′. Since g^{−1}g′ ∈ G, by Theorem 6, 4) g^{−1}g′(v_{0}) can not be an adjoined vertex with v_{0}. Hence g^{−1}g′(v_{0}) = v_{0} or the other end of the diagonal. In both cases g^{−1}g′ of the diagonal [v_{0}, w′] of F′ is the diagonal [v_{0}, w].

We may assume that a torsion free group Γ′ of Theorem 7 is a subgroup of G. For every g ∈ Γ′ and for every cube C ⊂ D_{0}, we have g(C) ∩ C = ∅. Indeed, if a vertices v and g(v) belong to C, then there exist an element q ∈ Γ of the form s_{i} or s_{i}s_{j} or s_{i}s_{j}s_{k} with q(v) = g(v). By 4) of Theorem 6 q is of a finite order. By 5) of Theorem 6 q = g. It contradicts to the fact that Γ′ is torsion free. Therefore Γ′ acts freely on Σ preserving the triangulation. For every simplex σ of the triangulation all vertices lie in different orbits of Γ′. Hence the orbit space L = Σ/Γ′ is a simplicial complex. By 2),3),5) of Theorem 6 the link of every vertex v ∈ Σ is homeomorphic to K, therefore Lk(w, L) is homeomorphic to K for all vertices v ∈ L.

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Department of Mathematics and Computer science, University of Miami, Coral Gables, FL 33124