Equivariant K-homology for some Coxeter groups

Rubén Sánchez-García^{∗}

April 2006

Abstract We obtain the equivariant K-homology of the classifying space \underline{E}W for W a right-angled or, more generally, an even Coxeter group. The key result is a formula for the relative Bredon homology of \underline{E}W in terms of Coxeter cells. Our calculations amount to the K-theory of the reduced C^{∗}-algebra of W, via the Baum-Connes assembly map.

1 Introduction

Consider a discrete group G. The Baum-Connes conjecture [1] identifies the Ktheory of the reduced C^{∗}-algebra of G, C_{r}^{∗}(G), with the equivariant K-homology of a certain classifying space associated to G. This space is called the classifying space for proper actions, written \underline{E}G. The conjecture states that a particular map between these two objects, called the assembly map,

μ_{i} : K_{i}^{G}(\underline{E}G) ⟶ K_{i}(C_{r}^{∗}(G)) i ≥ 0 ,

is an isomorphism. Here the left hand side is the equivariant K-homology of \underline{E}G and the right hand side is the K-theory of C_{r}^{∗}(G). The conjecture can be stated more generally [1, Conjecture 3.15].

The equivariant K-homology and the assembly map are usually defined in terms of Kasparov’s KK-theory. For a discrete group G, however, there is a more topological description due to Davis and Lück [4], and Joachim [10] in terms of spectra over the orbit category of G. We will keep in mind this topological viewpoint (cf. Mislin’s notes in [14]).

Part of the importance of this conjecture is due to the fact that it is related to many other relevant conjectures [14, §7]. Nevertheless, the conjecture itself allows the computation of the K-theory of C_{r}^{∗}(G) from the K^{G}-homology of \underline{E}G. In turn, this K-homology can be achieved by means of the Bredon homology of \underline{E}G, as we explain later.

In this article we focus our attention on (finitely generated) Coxeter groups. These groups are well-known in geometric group theory as groups generated by reflections (elements of order 2) and only subject to relations in the form (st)^{n} = 1. Coxeter groups have the Haagerup property [2] and therefore satisfy the Baum-Connes conjecture [8].

For W a Coxeter group, we consider a model of the classifying space \underline{E}W called the Davis complex. We obtain a formula for the relative Bredon homology of this space in terms of Coxeter cells (Theorem 5.2). From this, we can deduce the Bredon homology of \underline{E}W in same cases; for instance, for right-angled Coxeter groups and, more generally, even Coxeter groups (Theorem 7.1 and Corollary 7.2). The equivariant K-homology of \underline{E}W follows immediately. Since Coxeter groups satisfy Baum-Connes, our results also amount to the K-theory of the corresponding reduced C^{∗}-algebra.

∗Funded by the EPSRC and the School of Mathematics, University of Southampton

These results appear in the author’s PhD thesis [17, Chapter 5]. I would like to thank my PhD supervisor Ian Leary for his guidance through this research project, and particularly for suggesting the use of the Coxeter cell structure and relative Bredon homology.

2 Preliminaries

2.1 Classifying spaces

The classifying space \underline{E}G that appears in the Baum-Connes conjecture is a particular case of a more general construction.

Let G be a discrete group. A G-CW-complex is a CW-complex with a Gaction permuting the cells and such that if a cell is sent to itself, it is done by the identity map. Let 𝔉 be a non-empty family of subgroups of G closed under conjugation and passing to subgroups. A model for E_{𝔉}G is a G-CW-complex X such that (1) all cell stabilizers are in 𝔉; (2) for any G-CW-complex Y with all cell stabilizers in 𝔉, there is a G-map Y → X, unique up to G-homotopy equivalence. The last condition is equivalent to the statement that for each H ∈ 𝔉, the fixed point subcomplex X^{H} is contractible.

For the family with just the trivial subgroup, we obtain EG, a contractible free G-CW-complex whose quotient BG is the classifying space for principal G-bundles; when 𝔉 = 𝔉𝔦𝔫(G), the family of all finite subgroups of G, this is the definition of \underline{E}G.

It can be shown that general classifying spaces E_{𝔉}G always exists. They are clearly unique up to G-homotopy. See [1, §2] or [14] for more information and examples of \underline{E}G.

2.2 Bredon homology

Given a group G and a family 𝔉 of subgroups, we will write O_{𝔉}G for the orbit category. The objects are left cosets G/K, K ∈ 𝔉, and morphisms the Gmaps ϕ : G/K → G/L. Such a G-map is uniquely determined by its image ϕ(K) = gL, and we have g^{−1}Kg ⊂ L. Conversely, such g ∈ G defines a G-map, which will be written R_{g}.

A covariant (resp. contravariant) functor M: O_{𝔉}G → 𝔄𝔟 is called a left (resp. right) Bredon module. The category of left (resp. right) Bredon modules and natural transformations is written G-Mod_{𝔉} (resp. Mod_{𝔉}-G). It is an abelian category, and we can use homological algebra to define Bredon homology (see [14, pp. 7-10]). Nevertheless, we give now a practical definition.

Consider a G-CW-complex X, and write ℑ𝔰𝔬(X) for the family of isotropy subgroups {stab(x), x ∈ X}. Let 𝔉 be a family of subgroups of G containing ℑ𝔰𝔬(X), and M a left Bredon module. The Bredon homology groups H_{i}^{𝔉} (X; M) can be obtained as the homology of the following chain complex. Let {e_{α}} be orbit representatives of the d-cells (d ≥ 0) and write S_{α} for stab(e_{α}) ∈ 𝔉. Define

C_{d} = ⨁_{α} M (G/S_{α}) .

If g · e′ is a typical (d − 1)-cell in the boundary of e_{α} then g^{−1} · stab(e_{α}) · g ⊂ stab(e′), giving a G-map (write S′ for stab(e′))

R_{g} : G/S_{α} → G/S′ ,

which induces a homomorphism M(ϕ): M (G/S_{α}) → M (G/S′), usually written (R_{g})_{∗}. This yields a differential ∂_{d}: C_{d} → C_{d−1}, and the Bredon homology groups H_{i}^{𝔉} (X; M) correspond to the homology of (C_{∗}, ∂_{∗}). Observe that the definition is independent of the family 𝔉 as long as it contains ℑ𝔰𝔬(X).

The Bredon homology of a group G with coefficients in M ∈ G-Mod_{𝔉} can be defined in terms of a Tor functor ([14, Def. 3.12]). If 𝔉 is closed under conjugations and taking subgroups then

H_{i}^{𝔉}(G; M) ≅ H_{i}^{𝔉}(E_{𝔉}G; M) .

We are interested in the case X = \underline{E}G, 𝔉 = 𝔉𝔦𝔫(G) and M = R the complex representation ring, considered as a Bredon module as follows. On objects we set

R(G/K) = R_{ℂ}(K), K ∈ 𝔉𝔦𝔫(G)

the ring of complex representations of the finite group K (viewed just as an abelian group), and for a G-map ϕ = R_{g} : G/K → G/L, we have g^{−1}Kg ⊂ L so that R(ϕ) = (R_{g})_{∗} : R_{ℂ}(K) → R_{ℂ}(L) is given by induction from g^{−1}Kg to L after identifying R_{ℂ}(g^{−1}Kg) ≅ R_{ℂ}(K).

2.3 Equivariant K-homology

There is an equivariant version of K-homology, denoted K_{i}^{G}(−) and defined in [4] (see also [10]) using spaces and spectra over the orbit category of G. It was originally defined in [1] using Kasparov’s KK-theory. We will only recall the properties we need.

Equivariant K-theory satisfies Bott mod-2 periodicity, so we only consider K_{0}^{G} and K_{1}^{G}. For any subgroup H of G, we have

K_{i}^{G} (G/H) = K_{i} (C_{r}^{∗}(H)) ,

that is, its value at one-orbit spaces corresponds to the K-theory of the reduced C^{∗}-algebra of the typical stabilizer. If H is a finite subgroup then C_{r}^{∗}(H) = ℂH and

K_{i}^{G} (G/H) = K_{i}(ℂH) = \{ \begin{matrix}R_{ℂ}(H) & i = 0 , \\ 0 & i = 1 . \end{matrix}

This allows us to view K_{i}^{G}(−) as a Bredon module over O_{𝔉𝔦𝔫}G.

We can use an equivariant Atiyah-Hirzebruch spectral sequence to compute the K^{G}-homology of a proper G-CW-complex X from its Bredon homology (see [14, pp. 49-50] for details), as

E_{p,q}^{2} = H_{p}^{𝔉𝔦𝔫(G)} (X; K_{q}^{G}(−)) ⇒ K_{p+q}^{G} (X) .

In the simple case when Bredon homology concentrates at low degree, it coincides with the equivariant K-homology:

Proposition 2.1. Write H_{i} for H_{i}^{𝔉𝔦𝔫(G)}(X; R) and K_{i}^{G} for K_{i}^{G}(X). If H_{i} = 0 for i ≥ 2 then K_{0}^{G} = H_{0} and K_{1}^{G} = H_{1}.

Proof. The Atiyah-Hirzebruch spectral sequence collapses at the 2-page. ♢

3 Künneth formulas for Bredon homology

We will need Künneth formulas for (relative) Bredon homology. We devote this section to state such formulas, for the product of spaces and the direct product of groups. Theorem 3.1 holds in more generality (see [7]) but we state the result we need and sketch a direct proof; details can be found in [17, Chapter 3]. Results similar to those explained here are treated in [11].

Künneth formula for X ×Y

Let X be a G-CW-complex and Y a H-CW-complex. Let 𝔉 (resp. 𝔉′) be a family of subgroups of G (resp. of H) containing ℑ𝔰𝔬(X) (resp. ℑ𝔰𝔬(Y)). Then X ×Y is a (G×H)-CW-complex (with the compactly generated topology) and

ℑ𝔰𝔬(X ×Y) = ℑ𝔰𝔬(X) × ℑ𝔰𝔬(Y) ⊂ 𝔉 × 𝔉′ .

Moreover, the orbit category O_{𝔉×𝔉′}(G×H) is isomorphic to O_{𝔉}G × O_{𝔉′}H.

Given M ∈ G-Mod_{𝔉}, N ∈ H-Mod_{𝔉′} we define their tensor product over ℤ as the composition of the two functors

\xymatrix{M ⊗ N : O_{𝔉}G × O_{𝔉′}H \ar[r]^{M×N} & 𝔄𝔟×𝔄𝔟 \ar[r]^{⊗_{ℤ}} & 𝔄𝔟} ,

considered as a Bredon module over O_{𝔉×𝔉′}(G×H) ≅ O_{𝔉}G × O_{𝔉′}H. We can easily extend this tensor product to chain complexes of Bredon modules.

Theorem 3.1 (Künneth Formula for Bredon Homology). Consider X, Y, 𝔉, 𝔉′, M, N as above, with the property that M(G/K) and N(H/K′) are free for all K ∈ ℑ𝔰𝔬(X), K′ ∈ ℑ𝔰𝔬(Y). Then for every n ≥ 0, there is a split exact sequence

0 → ⨁_{i+j=n} (H_{i}^{𝔉}(X; M) ⊗_{ℤ} H_{j}^{𝔉′}(Y; N)) → H_{n}^{𝔉×𝔉′}(X ×Y; M ⊗ N)

→ ⨁_{i+j=n−1} Tor(H_{i}^{𝔉}(X; M), H_{j}^{𝔉′}(Y; N)) → 0 .

Proof. (sketch) Consider the chain complexes of abelian groups

D_{∗} = \underline{C_{∗}(X)} ⊗_{𝔉} M and D′_{∗} = \underline{C_{∗}(Y)} ⊗_{𝔉′} N ,

where \underline{C_{∗}(−)} is the chain complex of Bredon modules defined in [14, p. 10-11], and − ⊗_{𝔉} − is the tensor product defined in [14, p. 14]. Then, by Definition 3.13 in [14],

H_{i}(D_{∗}) = H_{i}^{𝔉}(X; M) , H_{j}(D′_{∗}) = H_{j}^{𝔉′}(Y; N) .

We now use the following lemmas, which follow from the definitions of \underline{C_{∗}(−)} and − ⊗_{𝔉} −.

Lemma 3.2. Given \underline{C_{∗}} a chain complex in Mod_{𝔉}-G, \underline{C′_{∗}} a chain complex in Mod_{𝔉′}-H and Bredon modules M ∈ G-Mod_{𝔉}, N ∈ H-Mod_{𝔉′}, the following chain complexes of abelian groups are isomorphic

(\underline{C_{∗}} ⊗_{𝔉} M) ⊗ (\underline{C′_{∗}} ⊗_{𝔉′} N) ≅ (\underline{C_{∗}} ⊗ \underline{C′_{∗}}) ⊗_{𝔉×𝔉′} (M ⊗ N).

Lemma 3.3. The following chain complexes in Mod_{𝔉×𝔉′}-(G×H) are isomorphic

\underline{C_{∗}(X ×Y)} ≅ \underline{C_{∗}(X)} ⊗ \underline{C_{∗}(Y)} .

By Lemma 3.2,

D_{∗} ⊗_{ℤ} D′_{∗} ≅ (\underline{C_{∗}(X)} ⊗ \underline{C_{∗}(Y)}) ⊗_{𝔉×𝔉′} (M ⊗ N)

and therefore, by Lemma 3.3,

H_{n}(D_{∗} ⊗_{ℤ} D′_{∗}) = H_{n}^{𝔉×𝔉′}(X ×Y; M ⊗ N) .

The result is now a consequence of the ordinary Künneth formula of chain complexes of abelian groups. Observe that the chain complexes D_{∗} and D′_{∗} are free since

D_{n} = \underline{C_{n}(X)} ⊗_{𝔉} M ≅ ⨁_{α} M(G/S_{α})

(see [14, p. 14-15]), and similarly for D′_{n}. ♢

There is also a Künneth formula for relative Bredon homology (see [14, p. 12] for definitions).

Theorem 3.4 (Künneth Formula for Relative Bredon Homology). Under the same hypothesis of Theorem 3.1, plus A ⊂ X a G-CW-subcomplex and B ⊂ Y a H-CW-subcomplex, we have that for every n ≥ 0 there is a split exact sequence

0 ⟶ ⨁_{i+j=n} (H_{i}^{𝔉}(X, A; M) ⊗_{ℤ} H_{j}^{𝔉′}(Y, B; N))

⟶ H_{n}^{𝔉×𝔉′}(X ×Y, A×Y ∪ X ×B; M ⊗ N)

⟶ ⨁_{i+j=n−1} Tor(H_{i}^{𝔉}(X, A; M), H_{j}^{𝔉′}(Y, B; N)) ⟶ 0 .

The proof is the same as the non-relative formula but using the chain complexes

\begin{matrix}D_{∗} = \underline{C_{∗}(X, A)} ⊗_{𝔉} M \\ D′_{∗} = \underline{C_{∗}(Y, B)} ⊗_{𝔉′} M \end{matrix}

and a relative version of Lemma 3.3.

Künneth formula for G×H

We will work in general with respect to a class of groups 𝔉 (formally, 𝔉 is a collection of groups closed under isomorphism, or a collection of isomorphism types of groups). Our main example will be 𝔉𝔦𝔫, the class of finite groups.

Given a specific group G, let us denote by 𝔉(G) the family of subgroups of G which are in 𝔉, and by H_{∗}^{𝔉}(−) the Bredon homology with respect to 𝔉(G). Suppose now that 𝔉 is closed under taking subgroups (note that it is always closed under conjugation). Then so is 𝔉(G) and we will write E_{𝔉}G for E_{𝔉(G)}G.

We want to know when E_{𝔉}G × E_{𝔉}H is a model for E_{𝔉}(G × H).

Proposition 3.5. Let 𝔉 be a class of groups which is closed under subgroups, finite direct products and homomorphic images. Then, for any groups G and H, and X and Y models for E_{𝔉}G respectively E_{𝔉}H, the space X × Y is a model for E_{𝔉}(G×H).

Proof. If (x, y) ∈ X × Y then

stab_{G×H}(x, y) = stab_{G}(x) × stab_{H}(y) ∈ 𝔉(G) × 𝔉(H) ⊂ 𝔉(G × H) .

Consider the projections π_{1} : G×H → G and π_{2} : G×H → H. If K ∈ 𝔉(G×H),

(X × Y)^{K} = X^{π_{1}(K)} × Y^{π_{2}(K)}

contractible since π_{1}(K) ∈ 𝔉(G) and π_{2}(K) ∈ 𝔉(H). ♢

Thus we can apply the Künneth formula (Theorem 3.1) to E_{𝔉}(G×H). We obtain Bredon homology groups with respect to the family 𝔉(G)×𝔉(H) instead of 𝔉(G×H), but both families contain the isotropy groups of E_{𝔉}G × E_{𝔉}H, so the Bredon homology groups are the same (cf. Section 2.2).

Theorem 3.6 (Künneth Formula for G×H). Let 𝔉 be a class of groups closed under taking subgroups, direct product and homomorphic images. For every n ≥ 0 there is a split exact sequence

0 → ⨁_{i+j=n} (H_{i}^{𝔉}(G; M) ⊗_{ℤ} H_{j}^{𝔉}(H; N)) → H_{n}^{𝔉}(G×H; M ⊗ N) →

⨁_{i+j=n−1} Tor(H_{i}^{𝔉}(G; M), H_{j}^{𝔉}(H; N)) → 0 .

Remark 1. The analogous result for relative Bredon homology also holds, by Theorem 3.4.

Application to proper actions and coefficients in the representation ring

The original motivation for stating a Künneth formula was the Bredon homology of G×H for proper actions (i.e. 𝔉 = 𝔉𝔦𝔫) and coefficients in the representation ring. We only need a result on the coefficients.

For finite groups P and Q, the representation ring R_{ℂ}(P ×Q) is isomorphic to R_{ℂ}(P) ⊗ R_{ℂ}(Q). The same is true at the level of Bredon coefficient systems. Given groups G and H, consider the left Bredon modules

R^{G} ∈ G-Mod_{𝔉𝔦𝔫(G)} , R^{H} ∈ H-Mod_{𝔉𝔦𝔫(H)} and R^{G×H} ∈ (G×H)-Mod_{𝔉𝔦𝔫(G×H)} .

Let us denote by R̃^{G×H} the restriction of R^{G×H} to 𝔉𝔦𝔫(G)×𝔉𝔦𝔫(H).

Proposition 3.7. These two Bredon modules are naturally isomorphic

R^{G} ⊗ R^{H} ≅ R̃^{G×H} .

Proof. Consider the isomorphism Θ_{P,Q} : R_{ℂ}(P ×Q) ≅ R_{ℂ}(P) ⊗ R_{ℂ}(Q) and the universal property of the tensor product to obtain a map α as

\xymatrix{R_{ℂ}(P)×R_{ℂ}(Q) \ar[r] \ar[dr]_{α} & R_{ℂ}(P) ⊗ R_{ℂ}(Q) \ar[d]_{≅}^{Θ_{P,Q}} \\  & R_{ℂ}(P ×Q) .}

That is, Θ_{P,Q}(ρ ⊗ τ) = α(ρ, τ), where the latter denotes the ordinary tensor product of representations and ρ ⊗ τ is their formal tensor product in R_{ℂ}(P) ⊗ R_{ℂ}(Q). Extend this map to a natural isomorphism Θ : R^{G} ⊗ R^{H} → R̃^{G×H}. It only remains to check naturality, that is, the commutativity of

\xymatrix{R_{ℂ}(P) ⊗ R_{ℂ}(Q) \ar[r]^{Θ_{P,Q}} \ar[d]_{(R_{g})_{∗}⊗(R_{h})_{∗}} & R_{ℂ}(P × Q) \ar[d]^{(R_{(g,h)})_{∗}} \\ R_{ℂ}(P′) ⊗ R_{ℂ}(Q′) \ar[r]^{Θ_{P′,Q′}} & R_{ℂ}(P′ × Q′) ,}

for any (G × H)-map R_{(g,h)} = R_{g} × R_{h}. To do this, one can compute both sides of the square and show that the two representations have the same character, using the following two observations. Denote by χ(−) the character of a representation. Since α(−, −) is the tensor product of representations we have χ (α(τ^{1}, τ^{2})) = χ(τ^{1}) · χ(τ^{2}). If R_{g} : G/K → G/L is a G-map, ρ ∈ R_{ℂ}(K) and t ∈ G then

χ ((R_{g})_{∗} ρ) (t) = \frac{1}{|K|} ∑_{∗} χ(ρ)(k) ,

where the sum ∗ is taken over the s ∈ G such that k = gsts^{−1}g^{−1} ∈ K. We leave the details to the reader. ♢

For an arbitrary group G, write H_{i}^{𝔉𝔦𝔫}(G; R) for H_{i}^{𝔉𝔦𝔫(G)}(G; R^{G}).

Corollary 3.8. For every n ≥ 0 there is a split exact sequence

0 → ⨁_{i+j=n} (H_{i}^{𝔉𝔦𝔫}(G; R) ⊗_{ℤ} H_{j}^{𝔉𝔦𝔫}(H; R)) → H_{n}^{𝔉𝔦𝔫}(G×H; R) →

⨁_{i+j=n−1} Tor(H_{i}^{𝔉𝔦𝔫}(G; R), H_{j}^{𝔉𝔦𝔫}(H; R)) → 0 .

Remark 2. Again, there is a similar statement for relative Bredon homology.

4 Coxeter groups and the Davis complex

We briefly recall the definitions and basic properties of Coxeter groups, and describe a model of the classifying space for proper actions. Most of the material is well-known and can be found in any book on the subject, as [9].

4.1 Coxeter groups

Suppose W is a group and S = {s_{1}, …, s_{N}} is a finite subset of elements of order 2 which generate W. Let m_{ij} denote the order of s_{i}s_{j}. Then 2 ≤ m_{ij} = m_{ji} ≤ ∞ if i ≠ j and m_{ii} = 1. We call the pair (W, S) a Coxeter system and W a Coxeter group if W admits a presentation

⟨ s_{1}, …, s_{N} | (s_{i}s_{j})^{m_{ij}} = 1 ⟩ .

We call N the rank of (W, S). The Coxeter diagram is the graph with vertex set S and one edge joining each pair {s_{i}, s_{j}}, i ≠ j, with label m_{ij}, and the conventions: if m_{ij} = 2 we omit the edge; if m_{ij} = 3 we omit the label. A Coxeter system is irreducible if its Coxeter diagram is connected. Every Coxeter group can be decomposed as a direct product of irreducible ones, corresponding to the connected components of its Coxeter diagram. All finite irreducible Coxeter systems have been classified; a list of their Coxeter diagrams can be found, for instance, in [9, p. 32].

Every Coxeter group can be realized as a group generated by reflections in ℝ^{N}. Namely, there is a faithful canonical representation W → GL_{N}(ℝ) (see [9, §5.3] for details). The generators in S correspond to reflections with respect to hyperplanes. These hyperplanes bound a chamber, which is a strict fundamental domain (examples below).

Coxeter cells

Suppose now that W is finite. Take a point x in the interior of the chamber. The orbit W x is a finite set of points in ℝ^{N}. Define the Coxeter cell C_{W} as the convex hull of W x. It is a complex polytope by definition.

Examples:

(1) Rank 0: W = {1} the trivial group and C_{W} is a point.

(2) Rank 1: W = C_{2} the cyclic group of order 2, ℝ^{N} = ℝ, the W -action consists on reflecting about the origin, and there are two chambers, (−∞, 0] and [0, +∞). Take a point x in the interior of, say, [0, +∞); then W x = {−x, x} and C_{W} = [−x, x], an interval.

(3) Rank 2: W = D_{n} dihedral group of order 2n, ℝ^{N} = ℝ^{2} and the canonical representation identifies W with the group generated by two reflections with respect to lines through the origin and mutual angle π/n. A chamber is the section between the two lines and the W -orbit W x is the vertex set of a 2n-gon (see Figure 1). Therefore, C_{W} is a 2n-gon.

(4) If W = W_{1} × W_{2} then C_{W} = C_{W_{1}} × C_{W_{2}}.

The Davis complex

It is a standard model of \underline{E}W introduced by M. Davis in [5]. Given T ⊂ S, the group W_{T} = ⟨T ⟩ generated by T is called a special subgroup. It can be shown that (W_{T} , T) is a Coxeter system, and that W_{T} ∩ W_{T′} = W_{T ∩T′}. The subset T is called spherical is W_{T} is finite. We will write S for the set {T ⊂ S | T is spherical}.

Figure 1: Coxeter cell for a dihedral group.

A spherical coset is a coset of a finite special subgroup, that is, wW_{T} for some w ∈ W, T ∈ S. Note that wW_{T} = w′W_{T′} if and only if T = T′ and w^{−1}w′ ∈ W_{T} (use the so-called deletion condition [9, §5.8]). Denote by W S the set of all spherical cosets in W, that is, the disjoint union

W S = ⋃_{T ∈S} W/W_{T} .

This is a partially ordered set (poset) by inclusion, and there is a natural W - action whose quotient space is S. The Davis complex Σ is defined as the geometric realization |W S| of the poset W S, that is, the simplicial complex with one n-simplex for each chain of length n + 1

w_{0}W_{T_{0}} ⊂ w_{1}W_{T_{1}} ⊂ … ⊂ w_{n}W_{T_{n}} , T_{i} ∈ S ,  (4.1)

and obvious identifications. The W -action on W S induces an action on Σ for which this is a proper space: the stabilizer of the simplex corresponding to the chain (4.1) is w_{0}W_{T_{0}}w_{0}^{−1}, finite. Moreover, Σ admits a CAT(0)-metric [16] and therefore it is a model of \underline{E}W, by the following result.

Proposition 4.1. If a finite group H acts by isometries on a CAT(0)-space X, then the fixed point subspace X^{H} is contractible.

Proof. It is enough to find a fixed point for H; if x, y ∈ X are fixed by H then so is the geodesic between x and y, from which the contractibility of X^{H} follows. The existence of a fixed point for H is a consequence of the Bruhat-Tits fixed point theorem [3, p. 157]. ♢

Examples.

(1) \underline{Finite groups.-} Suppose that W is a finite Coxeter group and C is its Coxeter cell, defined as the convex hull of certain orbit W x. Denote by F(C) the poset of faces of the convex polytope C. The correspondence

w ∈ W ↦ wx ∈ W x = vertex set of C

allows us to identify a subset of W with a subset of vertices of C. In fact, a subset of W corresponds to the vertex set of a face of C if and only if it is a coset of a special subgroup ([3, III]); see Figure 1 for an example. Hence, we have the isomorphism of posets W S ≅ F(C) and the Davis complex Σ = |W S| is the barycentric subdivision of the Coxeter cell.

(2) \underline{Triangle groups.-} These are the Coxeter groups of rank 3,

⟨ a, b, c | a^{2} = b^{2} = c^{2} = (ab)^{p} = (bc)^{q} = (ca)^{r} = 1 ⟩ .

Suppose that p, q, r ≠ ∞ and \frac{1}{p} + \frac{1}{q} + \frac{1}{r} ≤ 1. All subsets T ⊊ S are spherical, giving the poset (where arrows stand for inclusions) of Figure 2. It can be realized as the barycentric subdivision of an euclidean or hyperbolic triangle with interior angles \frac{π}{p}, \frac{π}{q} and \frac{π}{r}, and a, b and c acting as reflections through the corresponding sides. The whole model consists on a tessellation of the euclidean or hyperbolic space by these triangles.

Figure 2: Davis complex (quotient) for a triangle group.

A second definition of the Davis complex

There is an alternative description of the Davis complex in terms of Coxeter cells. Given a poset P and an element X ∈ P, we denote by P_{≤X} the subposet of elements in P less or equal to X. Consider a spherical subset T ⊂ S and an element w ∈ W.

Proposition 4.2. There is an isomorphism of posets (W S)_{≤wW_{T}} ≅ W_{T} S.

Proof. Firstly, observe that the poset (W S)_{≤wW_{T}} is equivalent to (W S)_{≤W_{T}} via the isomorphism induced by multiplication by w^{−1}. So it suffices to show that W S_{≤W_{T}} = W_{T} S. A standard element in the right hand side is a coset wW_{T′} with w ∈ W_{T} and T′ ⊂ T so we have W_{T′} ⊂ W_{T} and wW_{T′} ⊂ wW_{T} = W_{T}. On the other hand, if wW_{T′} ⊂ W_{T} then w = w · 1 ∈ W_{T}. Also, T′ ⊂ T: if s′ ∈ T′ then ws′ ∈ W_{T} so s′ ∈ W_{T} and, using the deletion condition, s′ ∈ T. ♢

Now, since W_{T} is a finite Coxeter group, we can identify the subcomplex

| (W S)_{≤wW_{T}} | = |W_{T} S|

with the barycentric subdivision of the associated Coxeter cell C_{W_{T}}. From this point of view, the poset W S is the union of subposets (W S)_{≤wW_{T}} with T spherical and w ∈ W/W_{T}. That is, Σ can be viewed as the union of Coxeter cells

⋃_{\begin{matrix}T ∈S \\ wW_{T} ∈W/W_{T} \end{matrix}} C_{wW_{T}}

where C_{wW_{T}} is the copy of C_{W_{T}} corresponding to the coset wW_{T}. This union is obviously not disjoint, and the inclusions and intersections among subposets are precisely the following.

Lemma 4.3. (i) wW_{T} ⊂ w′W_{T′} if and only if T ⊂ T′ and w^{−1}w′ ∈ W_{T′}.

(ii) wW_{T} ∩ w′W_{T′} = w_{0}W_{T ∩T′} if there is any w_{0}W_{T} ∈ wW_{T} ∩ w′W_{T′}, empty otherwise.

Proof. (i) One implication is obvious, the other uses the deletion condition in the fashion of the proof of Proposition 4.2.

(ii) Straightforward, since W_{T} ∩ W_{T′} = W_{T ∩T′} . ♢

Consequently, the intersection of two Coxeter cells is

C_{wW_{T}} ∩ C_{w′W_{T′}} = \{ \begin{matrix}C_{w_{0}W_{T ∩T′}} & if wW_{T} ∩ w′W_{T′} = w_{0}W_{T ∩T′} \\ ∅ & if wW_{T} ∩ w′W_{T′} = ∅ . \end{matrix}  (4.2)

Denote by ∂C_{W} the boundary of the Coxeter cell, that is, the topological boundary in the ambient space ℝ^{N}. We have the following explicit description.

Proposition 4.4.

∂C_{W} = ⋃_{\begin{matrix}T ⊊S \\ wW_{T} ∈W/W_{T} \end{matrix}} wC_{W_{T}}

Proof. C_{W} is a convex polytope whose faces correspond to cosets of special subgroups (Example on page 9). The barycentric subdivision sd(C_{W}) = |W S| is indeed a cone over its center (the vertex corresponding to the coset W_{S} = W). ♢

5 Relative Bredon homology of the Davis complex

The aim of this section is to deduce a formula for the relative Bredon homology of the n-skeleton of the Davis complex Σ with respect to the (n − 1)-skeleton, H_{∗}^{𝔉𝔦𝔫} (Σ_{n}, Σ_{n−1}), in terms of the Bredon homology of the Coxeter cells.

We will denote by Σ the Davis complex with the Coxeter cell decomposition explained on page 10 and by Σ′ the original definition as the nerve of W S. Note that both are W -homeomorphic spaces and Σ′ = sd(Σ) (barycentric subdivision) but Σ is not a W -CW-complex while Σ′ is (recall that in a G-CW-complex a cell is sent to itself only by the identity map).

As an example, see Figure 3: it is the tessellation of the euclidean plane induced by the triangle group Δ(2, 4, 4) together with the dual tessellation given by squares and octagons. The latter is Σ, the Davis complex given as union of Figure 3: Tessellation of the euclidean plane given by the triangle group Δ(2, 4, 4), and its dual tessellation.

Coxeter cells. The skeleton filtration correspond to Coxeter cells of rank 0 (points), rank 1 (intervals) and rank 2 (2n-gons).

Definition 5.1. For n ≥ −1 define Σ_{n} ⊂ Σ as the union of Coxeter cells corresponding to finite W_{T} with rank(T) ≤ n. That is,

Σ_{n} = ⋃_{\begin{matrix}T ∈S, |T |≤n \\ wW_{T} ∈W/W_{T} \end{matrix}} C_{wW_{T}} .

Then Σ_{−1} = ∅, Σ_{0} is a free orbit of points and Σ_{1} is indeed the Cayley graph of W with respect to S (see Figure 3 for an example or compare with Proposition 5.4).

We have that Σ_{n} is a W -subspace (w′C_{wW_{T}} = C_{w′wW_{T}}) and, since the dimension of C_{W_{T}} is rank(T), Σ_{n} is indeed the n-skeleton of Σ.

Our aim is to prove the following theorem.

Theorem 5.2. For any dimension n ≥ 0, any degree i and any Bredon coefficient system M, we have the isomorphism

H_{i}^{𝔉𝔦𝔫} (Σ_{n}, Σ_{n−1}; M) ≅ ⨁_{\begin{matrix}T ∈S \\ rank(T)=n \end{matrix}} H_{i}^{𝔉𝔦𝔫} (C_{W_{T}} , ∂C_{W_{T}}; M) .

Comments on the statement of the theorem:

(i) We have defined Bredon homology on G-CW-complexes so we implicitly assume that we take the barycentric subdivision on the spaces.

(ii) The Bredon homology on the left-hand side is with respect to the W - action and the family 𝔉𝔦𝔫(W). The Bredon homology on the right-hand side is with respect to the W_{T} -action and the family 𝔉𝔦𝔫(W_{T}) = 𝔄𝔩𝔩(W_{T}) (all subgroups), for each Coxeter cell.

Firstly, we state some properties of the Coxeter cells that we will use.

Lemma 5.3. (a) Suppose T ∈ S with rank(T) ≤ n. Then

C_{W_{T}} ∩ Σ_{n−1} = \{ \begin{matrix}C_{W_{T}} & if rank(T) < n , \\ ∂C_{W_{T}} & if rank(T) = n . \end{matrix}

(b) The intersection of any two different Coxeter cells in Σ_{n} is included in Σ_{n−1}.

(c) The Coxeter cell C_{W_{T}} is a W_{T} –subspace of Σ_{n} and stab_{W} (x) is contained in W_{T} for all x ∈ C_{W_{T}}.

Proof. (a) If rank(T) < n then C_{W_{T}} ⊂ Σ_{n−1}. If rank(T) = n then (Proposition 4.4)

∂C_{W_{T}} = ⋃_{\begin{matrix}T′⊊T \\ wW_{T′} ∈W_{T} /W_{T′} \end{matrix}} wC_{W_{T′}} ⊂ Σ_{n−1} .

On the other hand, if x ∈ C_{W_{T}} ∩ Σ_{n−1} then x ∈ C_{wW_{T′}} with rank(T′) ≤ n − 1 so x ∈ C_{W_{T}} ∩ wC_{W_{T′}} which, by equation (4.2), equals to either empty or w_{0}C_{W_{T ∩T′}} ⊂ ∂C_{W_{T}}.

(b) In general, the intersection of two different Coxeter cells of rank n and m is either empty or another Coxeter cell of rank strictly less than min{n, m} — see equation (4.2).

(c) The first part is obvious: wC_{W_{T}} = C_{wW_{T}} = C_{W_{T}} if w ∈ W_{T}. Suppose now that x ∈ C_{W_{T}} and wx = x. We want w ∈ W_{T}. Identify sd(C_{W_{T}}) = |W_{T} S|. Suppose that x belongs to a cell corresponding to the chain W_{T_{0}} ⊊ … ⊊ W_{T_{k}}, T_{k} ⊂ T. If w fixes x, it has to fix at least one of the vertices of the cell, i.e., there is an i such that wW_{T_{i}} = W_{T_{i}} so that w ∈ W_{T_{i}} ⊂ W_{T}. ♢

Secondly, we observe that the copies of a Coxeter cell in Σ admit an interpretation as induced spaces. If H is a subgroup of G and X is an H-space, the associated induced G-space is

Ind_{H}^{G}X = G ×_{H} X ,

the orbit space for the H-action h · (g, x) = (gh^{−1}, hx). The (left) G-action on the induced space is given by g · \overline{(k, x)} = \overline{(gk, x)}. This definition carries on to pairs of H-spaces.

Proposition 5.4. For each spherical T ⊂ S, there is a W -homeomorphism

⋃_{wW_{T} ∈W/W_{T}} C_{wW_{T}} ≅_{W} W ×_{W_{T}} C_{W_{T}} .

Proof. If x ∈ C_{wW_{T}} = w · C_{W_{T}}, write x = wx_{0} and send x to \overline{(w, x_{0})} ∈ W ×_{W_{T}} C_{W_{T}}. This is well-defined by equation (4.2): x = wx_{0} = w′y_{0} then there is w′′ ∈ W_{T} such that w = w′w′′. This defines a continuous W -map, bijective, with continuous inverse

\overline{(w, x)} ↦ w · x ∈ C_{wW_{T}} .  ♢

Consequently, we may write Σ_{n}, n ≥ 1, as a union of induced spaces

Σ_{n} ≅ ⋃_{|T |≤n} W ×_{W_{T}} C_{W_{T}} .

Next, we will need the following easy consequence of a relative Mayer-Vietoris sequence.

Proposition 5.5. Let X be a G-CW-complex and Y, A_{1}, …, A_{n} G-subcomple-xes such that X = A_{1} ∪ … ∪ A_{n} and A_{i} ∩ A_{j} ⊂ Y for all i ≠ j. Write H_{n}(−) for Bredon homology with some fixed coefficients and with respect to a family 𝔉 ⊃ ℑ𝔰𝔬(X) or, more generally, any G-homology theory. Then

H_{n}(X, Y) ≅ ⨁_{i=1}^{n} H_{n}(A_{i}, A_{i} ∩ Y) .

Proof. By induction on n. For n = 1 it is a tautology. For n > 1 call A = A_{1}, B = A_{2} ∪ … ∪ A_{n}, C = A ∩ Y and D = B ∩ Y. The relative Mayer-Vietoris of the CW-pairs (A, C) and (B, D) is

… → H_{i}(A∩B, C∩D) → H_{i}(A, C)⊕H_{i}(B, D) → H_{i}(A∪B, C∪D) → … (5.1)

Observe that

\begin{aligned}A ∪ B & = & X \\ C ∪ D & = & (A ∪ B) ∩ Y = Y \\ A ∩ B & ⊂ & Y \\ C ∩ D & = & A ∩ B ∩ Y = A ∩ B \end{aligned}

Therefore H_{i}(A ∩ B, C ∩ D) = 0 for all i and the sequence (5.1) gives isomorphisms

H_{i}(X, Y) ≅ H_{i}(A, C) ⊕ H_{i}(B, D)

Now apply induction to X′ = B, Y′ = B ∩ Y and the result follows. ♢

Finally, recall that if h_{∗}^{?} is an equivariant homology theory (see, for instance, [12, §1]), it has an induction structure. Hence, for a pair of H-spaces (X, A),

h_{n}^{G}(Ind_{H}^{G}(X, A)) ≅ h_{n}^{H}(X, A) .

Bredon homology has an induction structure [15]; in particular,

H_{n}^{𝔉𝔦𝔫(W)} (Ind_{W_{T}}^{W} (C_{W_{T}} , ∂C_{W_{T}})) ≅ H_{n}^{𝔉𝔦𝔫(W_{T})} (C_{W_{T}} , ∂C_{W_{T}}) .

Proof. (Theorem 5.2) Write H_{n}(−) for Bredon homology of proper G-spaces with coefficients in the representation ring. Let T_{1}, …, T_{m} be the spherical subsets of generators up to rank n. Define

A_{i} = W ×_{W_{T_{i}}} C_{W_{T_{i}}} 0 ≤ i ≤ m .

Then Σ_{n} = ⋃ A_{i} and A_{i} ∩ A_{j} ⊂ Σ_{n−1} for all i ≠ j, by Lemma 5.3(b). By Proposition 5.5

H_{n}(Σ_{n}, Σ_{n−1}) ≅ ⨁_{i=1}^{n} H_{n}(A_{i}, A_{i} ∩ Σ_{n−1}) .

If rank(T_{i}) < n then A_{i} ⊂ Σ_{n−1} and the corresponding term is zero. If rank(T_{i}) = n then A_{i} ∩ Σ_{n−1} = ∂A_{i} (Lemma 5.3(a)) and, by the induction structure,

H_{n}(A_{i}, ∂A_{i}) ≅ H_{n}(C_{W_{T_{i}}} , ∂C_{W_{T_{i}}}) .  ♢

Corollary 5.6. Write H_{i}(−) for Bredon homology with respect to finite subgroups and some fixed Bredon coefficient system. For each n ≥ 1, there is a long exact sequence

… → H_{i}^{𝔉𝔦𝔫}(Σ_{n−1}) → H_{i}^{𝔉𝔦𝔫}(Σ_{n}) → ⨁_{\begin{matrix}T ∈S \\ rank(T)=n \end{matrix}} H_{i}^{𝔉𝔦𝔫} (C_{W_{T}} , ∂C_{W_{T}}) → …

6 Relative Bredon homology of some Coxeter cells

As an application of Theorem 5.2, we compute the relative Bredon homology of (Σ_{n}, Σ_{n−1}), with coefficients in the representation ring, for the first cases n = 0, 1, 2. To do so, we need the Bredon homology of (C_{W_{T}} , ∂C_{W_{T}}) when T ⊂ S spherical with rank(T) = 0, 1, 2.

rank(T) = 0 T = ∅, W_{T} = {1} and C_{W_{T}} is a point. So the homology is R_{ℂ}({1}) ≅ ℤ at degree 0 and vanishes elsewhere.

rank(T) = 1 T = {s_{i}}, W_{T} is cyclic of order two and C_{W_{T}} is an interval. For the relative homology, we consider one 0-cell and one 1-cell, with stabilizers cyclic order two and trivial respectively (see Figure 4). The associated Bredon

Figure 4: Coxeter cell of rank 1

chain complex

\begin{aligned}0 & ⟶ & R_{ℂ}(\{1\}) & ⟶^{Ind} & R_{ℂ} (C_{2}) & ⟶ & 0 \\ & & 1 & ↦ & (1, 1) \end{aligned}

gives

H_{i} (C_{W_{T}} , ∂C_{W_{T}}; R) = \{ \begin{matrix}ℤ & i = 0 \\ 0 & i ≠ 0 \end{matrix} .  (6.1)

rank(T) = 2 T = {s_{i}, s_{j}}, W_{T} is dihedral of order 2n (n = m_{ij} ≠ ∞) and

Figure 5: Coxeter cell of rank 2

C_{W_{T}} is a 2n-gon. The orbit space is a sector of the polygon (see Figure 5) and the relative Bredon chain complex is

0 ⟶ R_{ℂ} ({1}) ⟶^{d_{2}} R_{ℂ} (C_{2}) ⊕ R_{ℂ} (C_{2}) ⟶^{d_{1}} R_{ℂ} (D_{n}) ⟶ 0 ,

where the differentials are given by induction among representation rings. Inducing the trivial representation always yields the regular one, so d_{2} in matrix form is

d_{2} ≡ ( 1 1 1 1 ) ∼ ( 1 0 0 0 ) ,

the right-hand side being its Smith normal form. For d_{1}, recall the character tables of the cyclic group of order two and dihedral group (with Coxeter generators a and b):

\begin{matrix}C_{2} & 1 & a \\ ρ_{1} & 1 & 1 \\ ρ_{2} & 1 & −1 \end{matrix}  \begin{matrix}D_{n} & (ab)^{k} & b(ab)^{k} \\ χ_{1} & 1 & 1 \\ χ_{2} & 1 & −1 \\ χ̂_{3} & (−1)^{k} & (−1)^{k} \\ χ̂_{4} & (−1)^{k} & (−1)^{k+1} \\ ϕ_{l} & 2 cos (2πlk/n) & 0 \end{matrix}

where 0 ≤ k ≤ n − 1, l varies from 1 to n/2 (n even) or (n − 1)/2 (n odd) and the hat denotes a character which only appears when n is even. The induced representations corresponding to the inclusions ⟨a⟩ ⊂ ⟨a, b⟩ and ⟨b⟩ ⊂ ⟨a, b⟩ are

Ind (ρ_{1}) = χ_{1} + χ̂_{3} + ∑ ϕ_{l} ,
Ind (ρ_{2}) = χ_{2} + χ̂_{4} + ∑ ϕ_{l} .

Consequently, the differential d_{1} is written in matrix form as

d_{1} ≡ ( \begin{matrix}1 & 0 & ˆ1 & ˆ0 & 1 & … & 1 \\ 0 & 1 & ˆ0 & ˆ1 & 1 & … & 1 \\ 1 & 0 & ˆ0 & ˆ1 & 1 & … & 1 \\ 0 & 1 & ˆ1 & ˆ0 & 1 & … & 1 \end{matrix} ) ,

which reduces by elementary operations to

( \begin{matrix}I_{3} & 0 \\ 0 & 0 \end{matrix} ) if n is even , ( \begin{matrix}I_{2} & 0 \\ 0 & 0 \end{matrix} ) if n is odd .

Therefore, the relative Bredon homology of a Coxeter cell of rank two and n = m_{ij} ≠ ∞ is

\begin{matrix}n even: & H_{0} = ℤ^{c(D_{n})−3} = ℤ^{n/2} , & & H_{i} = 0 ∀ i ≥ 1 ; \\ n odd: & H_{0} = ℤ^{c(D_{n})−2} = ℤ^{(n−1)/2} , & H_{1} = ℤ , & H_{i} = 0 ∀ i ≥ 2 . \end{matrix}

Here we write c(H) for the number of conjugacy classes in a finite group H, and ℤ^{n}, or sometimes n · ℤ, for ⊕_{i=1}^{n}ℤ.

We can now use Theorem 5.2 to deduce the lower-rank relative Bredon homology of Σ. We omit the family 𝔉𝔦𝔫 and coefficients R for clarity.

Proposition 6.1. Let (W, S) be a Coxeter system and Σ its associated Coxeter complex. Then

\begin{aligned}H_{i} (Σ_{0}, Σ_{−1}) & = & \{ \begin{matrix}ℤ & i = 0 \\ 0 & i ≠ 0 \end{matrix} . \\ H_{i} (Σ_{1}, Σ_{0}) & = & \{ \begin{matrix}|S| · ℤ & i = 0 \\ 0 & i ≠ 0 \end{matrix} . \end{aligned}

H_{0} (Σ_{2}, Σ_{1}) = \frac{1}{2} ( ∑_{even, i<j} m_{ij} + ∑_{odd, i<j}(m_{ij} − 1)) · ℤ ,

H_{1} (Σ_{2}, Σ_{1}) = |{m_{ij} odd, i < j}| · ℤ , H_{i} (Σ_{2}, Σ_{1}) = 0 (i ≥ 2) .

7 Bredon homology and K-homology of rightangled and even Coxeter groups

A Coxeter system (W, S) is right-angled if m_{ij} ∈ {2, ∞} for all i, j. More generally, a Coxeter system (W, S) is even if m_{ij} is even or infinite for all i, j. The only spherical subgroups are direct products of cyclic (order two) and dihedral, since any irreducible finite Coxeter group with more than two generators have at least one odd m_{ij}.

Suppose that W_{T} = D_{m_{1}} × … × D_{m_{r}} × (C_{2})^{k}, with m_{i} (1 ≤ i ≤ r) even numbers and k ≥ 0. We know that the relative Bredon homology of the Coxeter cells corresponding to dihedral D_{m_{i}} and C_{2} concentrates at degree 0 and

H_{0} = (m_{i}/2) · ℤ , respectively H_{0} = ℤ .

By the relative Künneth formula we have

H_{i} (C_{W_{T}} , ∂C_{W_{T}}) = \{ \begin{matrix}(m_{1} · . . . · m_{r}/2^{r}) · ℤ & i = 0 \\ 0 & i ≠ 0 \end{matrix} .

Alternatively, if T = {t_{1}, …, t_{n}} is a spherical subset of rank n,

H_{0} (C_{W_{T}} , ∂C_{W_{T}}) = ∏_{i<j} (m_{ij}/2) · ℤ ,  (7.1)

and 0 at any other degree. This expression is valid for any n ≥ 0, with the convention that an empty product equals 1.

Now we can compute the relative homology groups of the skeleton filtration of Σ. They concentrate at degree zero, since so does the homology of the Coxeter cells. We already know that (Proposition 6.1)

H_{0} (Σ_{0}, Σ_{−1}) = ℤ, H_{0} (Σ_{1}, Σ_{0}) = |S| · ℤ, H_{0} (Σ_{2}, Σ_{1}) = ∑_{\begin{matrix}i<j \\ m_{ij}≠∞ \end{matrix}} (m_{ij}/2) · ℤ .

In general, using Theorem 5.2 and equation (7.1), we have that H_{0} (Σ_{n}, Σ_{n−1}), n ≥ 0 is a free abelian group of rank

r(n) = ∑_{\begin{matrix}T ∈S \\ rank(T)=n \end{matrix}} ∏_{\begin{matrix}s_{i},s_{j}∈T \\ i<j \end{matrix}} m_{ij}/2 ,

with the convention that an empty sum is 0 and an empty product is 1. As before, the long exact sequences of (Σ_{n}, Σ_{n−1}) and induction on n gives

H_{i} (Σ_{n}) = H_{i} (Σ_{0}) = 0 for all i ≥ 1, and
H_{0} (Σ_{n}) = H_{0} (Σ_{n−1}) ⊕ H_{0} (Σ_{n}, Σ_{n−1}) .

Define

r = r(0) + … + r(|S|) = ∑_{T ∈S} ∏_{\begin{matrix}s_{i},s_{j}∈T \\ i<j \end{matrix}} m_{ij}/2 .

Theorem 7.1. The Bredon homology of an even Coxeter group is the free abelian group of rank r at degree 0 and vanishes at any other degree.

Corollary 7.2. The Bredon homology of a right-angled Coxeter group is the free abelian group of rank r = |S|, the number of spherical subsets, at degree 0 and vanishes at any other degree.

Finally, since the Bredon homology concentrates at degree 0, it coincides with the equivariant K-homology (Proposition 2.1). Also, the Baum-Connes conjecture holds for Coxeter groups, so we have the following corollary.

Corollary 7.3. If W is an even Coxeter group, the equivariant K-homology of \underline{E}W coincides with its Bredon homology at degree 0 and 1 respectively (given in the previous theorem). This corresponds to the topological K-theory of the reduced C^{∗}-algebra of W via the Baum-Connes assembly map.

8 Bredon homology of low-rank Coxeter groups

We briefly recall, for completeness, the Bredon homology of the Coxeter groups of rank up to three. If a group H is finite, a one-point space is a model of \underline{E}G so its Bredon homology reduces to the representation ring R_{ℂ}(H) at degree 0 and vanishes elsewhere, and so its equivariant K-homology. The first infinite Coxeter groups are the infinite dihedral

D_{∞} = ⟨a, b | a^{2} = b^{2} = 1⟩ ,

and the euclidean and hyperbolic triangle groups

⟨a, b, c | a^{2} = b^{2} = c^{2} = 1, (ab)^{p} = (bc)^{q} = (ca)^{r} = 1⟩ ,

where 2 ≤ p, q, r ≤ ∞ and 1/p + 1/q + 1/r ≤ 1. All cases can be done by direct computation from the correponding orbit spaces \underline{B}G ([17, Section 5.2]). The resulting Bredon homology concentrates at degree i ≤ 1 so it coincides with their K^{G}-homology. The results are the following (write H_{i}(G) for H_{i}^{𝔉𝔦𝔫} (\underline{E}G; R), n·ℤ for ⊕_{i=1}^{n}ℤ, and c(H) for the number of conjugacy classes on a finite group H).

H_{i} (D_{∞}) = \{ \begin{matrix}3 · ℤ & i = 0 \\ 0 & i ≠ 0 \end{matrix}

H_{i} (Δ(∞, ∞, ∞)) = \{ \begin{matrix}4 · ℤ & i = 0 \\ 0 & i ≠ 0 \end{matrix}

H_{i} (Δ(p, ∞, ∞)) = \{ \begin{matrix}(c(D_{p}) + 1) · ℤ & i = 0 \\ 0 & i ≠ 0 \end{matrix}

H_{i} (Δ(p, q, ∞)) = \{ \begin{matrix}(c(D_{p}) + c(D_{q}) − 2) · ℤ & i = 0 \\ 0 & i ≠ 0 \end{matrix}

H_{0} (Δ(p, q, r)) = \{ \begin{matrix}(c(D_{p}) + c(D_{q}) + c(D_{r}) − 4) · ℤ & p, q, r odd \\ (c(D_{p}) + c(D_{q}) + c(D_{r}) − 5) · ℤ & otherwise \end{matrix}

H_{1} (Δ(p, q, r)) = \{ \begin{matrix}ℤ & p, q and r odd \\ 0 & otherwise \end{matrix}

H_{i} (Δ(p, q, r)) = 0  for i ≥ 2 .

(Recall that c(D_{n}) is n/2 + 3 if n if even and (n − 1)/2 + 2 if n is odd.) Remark 3. Lück and Stamm [13] studied the equivariant K-homology of cocompact (= compact quotient) planar groups. Our groups above, except those triangle groups with p, q or r being ∞, are examples of cocompact planar groups so their K^{G}-homology also follow from Part (c) of Theorem 4.31 in their article.

Finally, observe that the results above agree with those of Section 7 when p, q and r are even or infinity.

9 A GAP routine for Coxeter groups

We have implemented a GAP [6] program to obtain the Bredon homology with coefficients in the representation ring of virtually any Coxeter group, from its Coxeter matrix.

The GAP routine works as follows. Firstly, we generate the cellular decomposition, cell stabilizers and boundary of the corresponding Davis complex. To do that, we have written functions to find the irreducible components of a Coxeter group and to decide whether the group it is finite (comparing each irreducible component with the finite Coxeter groups of the same rank). We generate a multidimensional list CHAINS such that CHAINS[i] is the list of all chains T_{1} < … < T_{i} of length i (< means strict inclusion) of subsets of {1, …, N} where N = rank(W, S). Note that a cell σ in the quotient space Σ/W corresponding to {T_{1} < … < T_{n}} has dimension n − 1, stabilizer W_{T_{1}} and boundary

∂σ = ∑_{k=1}^{n}(−1)^{k}{T_{1} < … < T̂_{i} < … T_{n}} .

Secondly, we use this information as initial data to another procedure which computes the Bredon homology with coefficients in the representation ring of a finite proper G-CW-complex. Note that, however, the computing time increases exponentially with the number of generators.

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