FINITE K(π, 1)’S FOR ARTIN GROUPS

Ruth Charney^{(∗)} and Michael W. Davis^{(∗)}

To Bill Browder for his sixtieth birthday

Introduction

It is not so widely known that there is a beautiful and simple description of a certain finite CW -complex Z which is a K(π, 1) for the braid group on n+1 strands. The complex Z is obtained by identifying certain faces of an n-dimensional convex polytope called a “permutohedron”. The resulting CW complex has exactly one kcell for each k-element subset of {1, …, n}. We are not sure who first discovered this complex, but we first heard it described by C. Squirer in the mid 1980’s and later by K. Tatsuoka [T]. It has also been known to J. Milgram for some time (a closely related construction appears already in [Mi]). Other references include [FSV], [P], and [S1]. The purpose of this paper is to give the details of the construction of this complex and its generalizations to other Artin groups.

It is a well-known fact that an Eilenberg-MacLane space for the pure braid group on n + 1 strands is the set of points in ℂ^{n+1} with all coordinates distinct. This space is the complement of an arrangement of hyperplanes in ℂ^{n+1} associated to the action of the symmetric group on n + 1 letters.

In [S], Salvetti showed how any hyperplane complement, obtained by complexifying a real arrangement, was homotopy equivalent to a certain cell complex. In the case at hand, this complex is a union of n-dimensional permutahedra. The symmetric group acts naturally and freely on it. Taking the quotient by the symmetric group, we obtain the complex Z.

There are two ingredients in the above program. The first is the fact that the hyperplane complement in ℂ^{n+1} is a K(π, 1). The second is Salvetti’s identification of the hyperplane complement with a certain cell complex.

Suppose that W is a Coxeter group with fundamental generating set S. In the same way as the braid groups on n+1 strands is associated to the symmetric group on n + 1 letters, one can associate to (W, S) an “Artin group” (or “generalized braid group”) A_{W}. We would like to carry out a similar program to obtain a nice K(π, 1)-complex for A_{W}.

As in [V] one can always find a representation of W as a linear reflection group on a real vector space V so that W acts properly on a certain nonempty, W -stable,

Typeset by AMS-TEX

(∗) Both authors were partially supported by NSF Grant DMS 9208071

convex, open set I (the interior of the “Tits cone”). When W is finite, I = V. Inside V ⊗ ℂ one has the domain V + iI. Consider the hyperplane complement

M = [(V + iI) − ∪ reflection hyperplanes]/W .

As for the first ingredient in the program, it is known that π_{1}(M) = A_{W} (see [CD1; Corollary 3.2.4] or [L]). In the case where W is finite, Deligne proved in [D] that M is a K(A_{W} , 1). In [CD1] the authors proved that the same result holds for many infinite Coxeter groups. (A precise statement can be found in §1.4, below.) Thus, this paper is a continuation of [CD1].

As explained in §1 and §2, the second ingredient (Salvetti’s complex) works for all W. Specifically, we show in Theorem 1.4.1 and Corollary 2.2.3, below, that M is homotopy equivalent to a certain finite CW -complex Z_{W}. The complex Z_{W} has one cell of dimension k for each k-element subset T of S such that the subgroup generated by T is finite. From this it is easy to compute the cohomological dimension of A_{W} (Corollary 1.4.2) and its Euler characteristic (Corollary 2.2.5) at least in the cases where M is known to be a K(π, 1)-space.

In §3, we consider the case where W is “right-angled”, that is, the product of any two generators has order either 2 or ∞. The first ingredient holds in this case. In Theorem 3.1.1, we show that Z_{W} with its natural piecewise Euclidean metric is nonpositively curved. Thus, in the right-angled case, A_{W} is a “semihyperbolic group” in the sense of [AB]. (This was previously proved by Hermiller and Meier in [HM] using combinatorial methods.) In this case, it is an easy matter to calculate the cohomology ring of A_{W} and we do so in Theorem 3.2.4.

After completing this paper we learned of Salvetti’s recent paper [S2], essentially on the same topic as this paper, at least in the case where W is finite.

1. Basic Definitions and Constructions

1.1. Coxeter groups and Artin groups.

A Coxeter matrix on a finite set S is an S by S symmetric matrix (m_{ss′}) such that each entry is a positive integer or ∞ and such that m_{ss′} = 1 if s = s′ and m_{ss′} ≥ 2 if s ≠ s′. Associated to (m_{ss′}) there is a group W defined by the presentation:

W = ⟨S | (ss′)^{m_{ss′}} = 1⟩ .

The natural map S → W is an injection ([B,Ch. V, §4.3, Prop. 4]) and henceforth, we identify S with its image in W. The pair (W, S) is a Coxeter system and W is a Coxeter group. If T is any subset of S then the subgroup W_{T}, generated by T, is called a special subgroup. It is known ([B; ch. IV, §1.8, Théoréme 2(i)]) that (W_{T} , T) is the Coxeter system associated to the restriction of the Coxeter matrix to T.

For each s ∈ S introduce a symbol x_{s} and let X = {x_{s}}_{s∈S}. Associated to a Coxeter matrix (m_{ss′}) there is also an Artin group, denoted A_{W}, and defined by the presentation

A_{W} = ⟨X | prod (x_{s}, x_{s′}, m_{ss′}) = prod (x_{s′}, x_{s}; m_{ss′})⟩

where prod (x, y; m) denotes the word xyx … of length m. We say that A_{W} is of finite type if W is a finite group.

If W is the symmetric group on n letters, then A_{W} is the braid groups on n strands. Thus, Artin groups are sometimes called “generalized braid groups”.

1.2. Posets and simplicial complexes associated to a Coxeter system.

Let (W, S) be a Coxeter system. Associated to (W, S) there are the following three posets:

S^{f} = {T ⊆ S | W_{T} is finite}
W S^{f} = {wW_{T} | w ∈ W, T ∈ S^{f}} ,
W × S^{f} .

The sets S^{f} and W S^{f} are partially ordered by inclusion. The partial order on W S^{f} is given explicitly as follows. If wW_{T} and w′W_{T′} are elements of W S^{f}, then wW_{T} < w′W_{T′} if and only if the following two conditions hold:

(1) T < T′,

(2) w^{−1}w′ ∈ W_{T′}. The partial ordering on W × S^{f} is defined as follows. If (w, T) and (w′, T′) are elements of W × S^{f}, then (w, T) < (w′, T′) if and only if conditions (1), (2) and the following hold:

(3) for all t ∈ T, ℓ(w^{−1}w′) < ℓ(tw^{−1}w′). (Here ℓ denotes word length with respect to the generating set S.) The geometric meaning of (3) will be given in §1. The natural map W × S^{f} → W S^{f} defined by (w, T) → wW_{T} is obviously order-preserving.

The posets S^{f} and W S^{f} appear in [CD1]. The poset W × S^{f} is one of the main objects of study in this paper.

For the moment, suppose that P is an arbitrary poset. Given p ∈ P, define a sub-poset, P_{≥p} = {x ∈ P | x ≥ p}. The sub-posets P_{≤p}, P_{>p} and P_{<p} are defined similarly. The poset P^{∘p}, called the dual of P, is equal to P as a set but with the order relations reversed. The derived complex of P, denoted by P′, is the set of finite chains in P, partially ordered by inclusion. It is an abstract simplicial complex.

The geometric realizations of (S^{f})′, (W S^{f})′, and (W ×S^{f})′ are denoted, respectively, by

\begin{aligned}K_{W} & = |(S^{f})′| \\ Σ_{W} & = |(W S^{f})′| \\ Σ̃_{W} & = |(W × S^{f})′| \end{aligned}

When there is no ambiguity, we shall omit the subscript “W ” from our notation. Following [CD1] we call Σ the modified Coxeter complex of W. Here Σ̃ will be called the Salvetti complex of W. We note that W acts naturally and simplicially on Σ and on Σ̃.

There is a projection map π: W S^{f} → S^{f} defined by wW_{T} → T and an embedding i: S^{f} → W S^{f} defined by T → W_{T}. These induce simplicial maps Σ → K and K → Σ which we again denote by π and i. We can identify K with its image i(K). The map π is then a retraction; moreover, it induces a simplicial isomorphism

Σ/W ≅ K .

The group W acts freely on Σ̃. The orbit space will be denoted by Z_{W} (or simply by Z), i.e.,

Σ̃/W = Z_{W} .

It is our primary object of study.

1.3. Sectors.

Let R denote the set of conjugates of S in W. An element of R is called a reflection. Given r ∈ R, its fixed points set on Σ is denoted by Σ^{r} and called a wall of Σ. The space Σ − Σ^{r} has two connected components, which are interchanged by r. Each such component is called a half-space. The half-space containing the interior of K (where K is regarded as a subcomplex of Σ) is called positive and denoted H_{+}^{r}.

Let T be a subset of S. Put

R_{T} = R ∩ W_{T} .

The components of

Σ − ⋃_{r∈R_{T}} Σ^{r}

are permuted freely and transitively by W_{T}. Each such component is called a W_{T} -sector of Σ. Such a sector is an intersection of half-spaces. The W_{T} -sector containing the interior of K is called positive. It is the intersection of the positive half-spaces H_{+}^{t}, t ∈ T.

If Λ is a W_{T} -sector then we shall also want to call its translate wΛ by an element w of W a “sector”. We note that wΛ is a component of

Σ − ⋃_{r∈wR_{T} w^{−1}} Σ^{r} .

We shall call it a wW_{T} w^{−1}-sector.

1.4. The spaces Q_{W} and Y_{W}.

Put

Y_{W} = Σ × Σ − ⋃_{r∈R} Σ^{r} × Σ^{r}

and

Q_{W} = Y_{W} /W .

As usual, we shall often omit the subscript W from our notation.

The following facts are proved in [CD1].

(i) Y is W -equivariantly homotopy equivalent to the complex hyperplane complement associated to any representation of W as a real linear reflection group ([CD1, Corollary 2.2.5]).

(ii) π_{1}(Q) = A_{W} ([CD1, Corollary 3.2.4]). (iii) π_{1}(Y) = P A_{W}, where P A_{W} denotes the kernel of the natural map A_{W} →

W. (P A_{W} is the “pure Artin group”.) The main conjecture of [CD1], then becomes the following.

Main Conjecture. Q_{W} is an Eilenberg-MacLane space K(A_{W} , 1).

The poset S_{>ϕ}^{f} is an abstract simplicial complex. Let K_{0} denote its geometric realization. (Thus, K is the cone on the barycentric subdivision of K_{0}.)

A simplicial complex is said to be a flag complex if any set of vertices which are pairwise joined by edges span a simplex. In [CD1], it is proved that the Main Conjecture holds under either of the following two hypotheses: (A) K_{0} is a flag complex, or (B) dim K_{0} ≤ 1. For example, (A) holds if W is finite.

The main result of this section is the following:

Theorem 1.4.1. Q_{W} is homotopy equivalent to Z_{W}.

Before giving a proof in the next subsection, let us deduce the corollary below. Put n_{W} = max{ Card (T) | T ∈ S^{f}}. In other words, n_{W} is the common

dimension of the simplicial complexes, K, Σ, Σ̃, and Z.

Corollary 1.4.2. Suppose the Main Conjecture holds for (W, S). Set n = n_{W}. Then the following statements are true.

(i) A_{W} has a finite, n-dimensional K(π, 1)-space, namely Z_{W}. (ii) A_{W} is of type F P. (iii) The cohomological dimension of A_{W} is n.

Proof. If the Main Conjecture holds, (i) follows immediately from Theorem 1.4.1 and (ii) follows immediately from (i).

Also by (i) the cohomological dimension of A_{W} is ≤ n. Suppose T ∈ S^{f} has n elements. By [Br], H^{n}(P A_{W_{T}}; ℤ) ≠ 0; hence, the cohomological dimension of A_{W} is ≥ n so (iii) holds. □

1.5. An open cover of Y.

The vertices of K are naturally indexed by the elements of S^{f}. For each T ∈ S^{f} let v_{T} denote the corresponding vertex of K. Similarly, the vertices of Σ are naturally indexed by W S^{f}. For each wW_{T} ∈ W S^{f}, let wv_{T} denote the corresponding vertex of Σ. Let Star (wv_{T}) denote the open star of wv_{T} in Σ. Consider the open cover of an arbitrary simplicial complex X by open stars of vertices. Of course, the nerve of this open cover is just X. Applying this fact in the case X = Σ gives the following lemma.

Lemma 1.5.1. Suppose w_{i}W_{T_{i}}, 0 ≤ i ≤ k, are distinct elements in W S^{f}. Then

Star (w_{0}v_{T_{0}}) ∩ ⋯ ∩ Star (w_{k}v_{T_{k}}) ≠ ∅

if and only if {w_{0}W_{T_{0}}, …, w_{k}W_{T_{k}}} is a chain in W S^{f}.

For each (w, T) ∈ W × S^{f}, let Sec (w, T) denotes the open wW_{T} w^{−1}-sector of Σ which contains the interior of wK. Define a subset U(w, T) of Σ × Σ by

U(w, T) = Sec (w, T) × Star (wv_{T}) .

Clearly, Σ^{r} ∩ Star (wv_{T}) ≠ ∅ if and only if r ∈ wR_{T} w^{−1}. Also, for all r ∈ wR_{T} w^{−1}, we have Sec (w, T) ∩ Σ^{r} = ∅. Hence, U(w, T) ∩ (Σ^{r} × Σ^{r}) = ∅ for all r in R. That is to say, U(w, T) is an open subset of Y. Moreover, U = {U(w, T)}_{(w,T)∈W ×S^{f}} covers Y.

Lemma 1.5.2.

(i) If U(w, T) = U(w′, T′), then (w, T) = (w′, T′). (ii) Suppose (w_{i}, T_{i}), 0 ≤ i ≤ k, are distinct elements in W × S^{f}. Then

U(w_{0}, T_{0}) ∩ ⋯ ∩ U(w_{k}, T_{k}) ≠ ∅ if and only if {(w_{0}, T_{0}), … (w_{k}, T_{k})} is a chain in W × S^{f}.

Proof. (i) Suppose U(w, T) = U(w′, T′). Then Star (wv_{T}) = Star (w′v_{T′}) and hence, wW_{T} = w′W_{T′}. It follows that T = T′ and that w^{−1}w′ ∈ W_{T}. The condition that Sec (w, T) = Sec (w′, T) is equivalent to Sec (1, T) = Sec (w^{−1}w′, T). Thus, w^{−1}w′ lies in W_{T} and the interior of w^{−1}w′K lies in the positive W_{T} -sector. Since W_{T} acts freely on the set of W_{T} -sectors, this forces, w^{−1}w′ = 1, i.e., w′ = w.

(ii) Suppose (w, T) and (w′, T′) are such that wW_{T} < w′W_{T′}. We claim that the following statements are then equivalent:

(a) (w, T) < (w′, T′),

(b) for all t ∈ T, ℓ(w^{−1}w′) < ℓ(tw^{−1}w′),

(c) Sec (w, T) ⊃ Sec (w′, T′),

(d) Sec (w, T) ∩ Sec (w′, T′) ≠ ∅. Condition (b) is just condition (3) of §1.2. Thus, (a) and (b) are equivalent. Given r ∈ R and u ∈ W, the condition that ℓ(u) < ℓ(ru) means that the interior of uK lies in the positive half-space for r. Thus, (b) means that the interior of w^{−1}w′K lies in the positive W_{T} -sector. By hypothesis, T < T′ so every W_{T′}-sector is contained some W_{T} -sector. Since w^{−1}w′ ∈ W_{T′}, we see that (b) is equivalent to the condition that Sec (1, T) ⊃ Sec (w^{−1}w′, T′) which is equivalent to (c). Obviously (c)⇒(d). On the other hand, since we are assuming T < T′ and w^{−1}w′ ∈ W_{T′}, we see that the intersection Sec (1, T) ∩ Sec (w^{−1}w′, T′) can be nonempty only if Sec (w^{−1}w′, T′) ⊂ Sec (1, T). Thus, (d)⇒(c). Statement (ii) of the lemma now follows from the previous lemma together with the equivalence of (a) and (d).

Proof of Theorem 1.4.1. Consider the open cover U = {U(w, T)}, of Y, indexed by the elements of W × S^{f}. Let σ = {(w_{0}, T_{0}), …, (w_{k}, T_{k})} be a set of distinct elements of W ×S^{f}, and put U_{σ} = U(w_{0}, T_{0})∩… U(w_{k}, T_{k}). By the previous lemma, U_{σ} is nonempty if and only if σ is a simplex in the derived complex (W × S^{f})′.

Moreover, if this is the case, then after renumbering we may assume that (w_{0}, T_{0}) < ⋯ < (w_{k}, T_{k}). It follows from the proof of the previous lemma (the equivalence of (c) and (d)) that

U_{σ} = Sec (w_{k}, T_{k}) × Star (p(σ))

where p: Σ̃ → Σ is the natural projection. It is shown in Lemma 2.2.6 of [CD1] that for T ∈ S^{f}, each W_{T} -sector of Σ is homotopy equivalent to K_{W_{T}}. In particular, U_{σ} is contractible. Hence, the nerve of U is Σ̃ and each nonempty intersection is contractible. It follows that the spaces Σ̃ and Y are homotopy equivalent. The group W acts freely on both spaces. We leave it as an exercise for the reader to construct a W -equivariant embedding Σ̃ → Y such that Y equivariantly deformation retracts onto Σ̃. Taking quotients by W we get the desired result: Z is homotopy equivalent to Q. □

2. A cell structure on Z

2.1. Coxeter cells.

In this subsection only, the Coxeter group W will be required to be finite. Thus, S^{f} will be the poset of all subsets of S. We shall describe a certain convex polytope P_{W}, called a “Coxeter cell”, such that the poset of faces of P_{W} is isomorphic to W S^{f}. Some of this material is also described in [CD2; §6].

Associated to (W, S) there is an (essentially unique) representation of W on ℝ^{n}, n = Card (S), as an orthogonal linear reflection group ([B; Ch. V, §4]). Each element of S acts on ℝ^{n} as an orthogonal reflection across a hyperplane. A “fundamental chamber” C is a simplicial cone bounded by the hyperplanes corresponding to the elements of S. To each function x: S → (0, ∞) there is a unique point, which we will also denote by x, in the interior of C such that the distance from x to the hyperplane fixed by s is x(s). Explicitly, if u_{s} is the outward-pointing unit normal vector to the hyperplane fixed by s, then x is the point defined by: x · u_{s} = −x(s), s ∈ S.

Definition 2.1.1. The Coxeter cell P_{W} associated to (W, S) and x is the convex hull of the W -orbit of x.

Examples 2.1.2.. (i) If W = ℤ/2, then P_{W} is an interval [−x(s), x(s)]. (ii) If W is the dihedral group of order 2m, then P_{W} is a 2m-gon. (iii) If (W, S) is a direct product of two Coxeter systems, (W, S) = (W_{1} ×

W_{2}, S_{1}∐S_{2}), then P_{W} is isometric to P_{W_{1}} × P_{W_{2}}.

(iv) In particular, if W ≅ (ℤ/2)^{n} and x is the constant function, then P_{W} is an n-cube.

Lemma 2.1.3. Suppose W is a finite Coxeter group. The poset of faces of the Coxeter cell P_{W} is isomorphic to W S^{f}. In particular, for T ⊂ S, the convex hull of the W_{T} -orbit of x is a face of P_{W} and this face is isomorphic to P_{W_{T}}.

Proof. Let e_{s} be a vector on the extremal ray of C which is opposite to the face fixed by s. Consider the linear form φ or ℝ^{n} defined by v → v · e_{s}. Put c = x · e_{s}. We claim that c is the maximum value of φ on P_{W}. Indeed, since C is a Dirichlet fundamental domain for the action of W on ℝ^{n} any point of C is at least as close to x as it is to any other point in the orbit of x. In particular, |wx−e_{s}|^{2} ≥ |x−e_{s}|^{2} for all w ∈ W. But this implies wx · e_{s} ≤ x · e_{s} for all w ∈ W and hence, that the maximum value of φ is attained at x. Let T = S − {s}. The affine hyperplane φ(v) = c contains the orbit of x under the subgroup W_{T} and is spanned by this orbit. It follows that φ(v) = c is a supporting hyperplane of P_{W} and that the convex hull of W_{T} x is a codimension one face of P_{W}.. Letting s vary over S we obtain all supporting hyperplanes containing the vertex x. Replacing x by wx and e_{s} by we_{s} we obtain in this way a description of all the supporting hyperplanes of P_{W}. The lemma follows easily. □

Remarks 2.1.4. (i) Associated to any convex polytope there is a dual polytope. The boundary complex of the dual polytope to P_{W} is called the Coxeter complex of W. The Coxeter complex is an (n − 1)-dimensional simplicial complex. It is combinatorically isomorphic to the triangulation of S^{n−1} whose spherical (n − 1)- simplices are the intersections S^{n−1} ∩ wC, w ∈ W.

(ii) Since Σ_{W} is the geometric realization of (W S^{f})′, we see that Σ_{W} can be identified with the barycentric subdivision of P_{W}.

2.2. Cellulations of Σ and Σ̃.

We return to the general situation where W can be infinite.

For each wW_{T} ∈ W S^{f} we have

(W S^{f})_{≤wW_{T}} ≅ W_{T} S_{≤T}^{f} .

Hence, the geometric realization of the derived complex of (W S^{f})_{≤wW_{T}} is a subcomplex of Σ_{W} isomorphic to Σ_{W_{T}}. By Lemma 2.1.3 and Remark 2.1.4(ii) we can identify this subcomplex with the Coxeter cell P_{W_{T}}. Thus, Σ_{W} is naturally cellulated by Coxeter cells. This gives Σ_{W} the structure of a convex cell complex: the associated poset of cells is W S^{f}.

Similarly, for each (w, T) ∈ W × S^{f}, we have

\begin{aligned}(W × S^{f})_{≤(w,T)} & ≅ (W_{T} × S_{≤T}^{f})_{≤(1,T)} \\ & ≅ W_{T} S_{≤T}^{f} \end{aligned}

where the second isomorphism is basically the observation that the projection W × S^{f} → W S^{f} restricts to an isomorphism

(W × S^{f})_{≤(w,T)} ≅ (W S^{f})_{≤wW_{T}} .

Hence, Σ̃_{W} also has a cellulation by Coxeter cells which projects to the cellulation on Σ_{W} described in the previous paragraph. We state this as the following lemma.

Lemma 2.2.1. Σ̃_{W} is cellulated by Coxeter cells: there is one cell for each element of W × S^{f}.

Remark 2.2.2. This cellulation of Σ̃ does not give it the structure of a convex cell complex in the strictest sense: the intersection of two cells need not be a common face of both, rather it is a union of such faces. For example, if W = ℤ/2, then Σ̃ is a circle, cellulated into two intervals.

Since the W -action on Σ̃_{W} is obviously cellular, we get the following corollary, which can be considered the main result of this paper.

Corollary 2.2.3. Z_{W} has the structure of a CW -complex: there is one cell of dimension Card (T) for each T ∈ S^{f}.

Corollary 2.2.4. The Euler characteristic, χ(Z_{W}) is given by the formula:

\begin{aligned}χ(Z_{W}) & = ∑_{T ∈S^{f}}(−1)^{Card (T)} \\ & = 1 − χ(K_{0}) . \end{aligned}

(Here K_{0} is as in §1.4: it is the geometric realization of the simplicial complex S_{>∅}^{f}.)

Proof. The first equation is immediate from the previous corollary. To see the second, note that the dimension of the simplex of K_{0} corresponding to T ∈ S_{>∅}^{f} is Card (T) − 1. Hence

χ(K_{0}) = − ∑_{T ∈S_{>∅}^{f}}(−1)^{Card (T)} .

Corollary 2.2.5. If the Main Conjecture holds for (W, S), then the Euler characteristic of the Artin group A_{W} is given by the same formula:

χ(A_{W}) = 1 − χ(K_{0}) .

Lemma 2.2.6. Suppose that (W, S) is the direct product of two Coxeter systems: (W, S) = (W_{1} × W_{2}, S_{1}∐S_{2}). Then

(i) A_{W} = A_{W_{1}} × A_{W_{2}}, (ii) Σ_{W} = Σ_{W_{1}} × Σ_{W_{2}}, (iii) Σ̃_{W} = Σ̃_{W_{1}} × Σ̃_{W_{2}}, (iv) Z_{W} = Z_{W_{1}} × Z_{W_{2}}.

Proof. Clear.

Corollary 2.2.7. Suppose W = (ℤ/2)^{n}. Then (i) A_{W} ≅ ℤ^{n}, (ii) Σ_{W} (= P_{W}) is an n-cube, (iii) Σ̃_{W} is an n torus (cellulated by 2^{n} n-cubes), and (iv) Z_{W} is an n-torus (formed by identifying opposite faces of a single n-cube in

the standard fashion).

2.3. Links.

We have just explained how the space Σ̃ is cellulated by Coxeter cells. The barycentric subdivision of this cell structure gives Σ̃ its natural simplicial structure discussed in §1.

Let (w, T) ∈ W × S^{f}. Then (w, T) corresponds to a vertex v in the simplicial structure on Σ̃. This vertex is the barycenter of a unique Coxeter cell σ (of dimension Card (T)). Any top-dimensional simplex in the barycentric subdivision of σ has v as its maximal vertex. The link of such a simplex in the simplicial structure on Σ̃ is the geometric realization of the derived complex of (W × S^{f})_{>(w,T)}. This can also be thought of as the barycentric subdivision of the link of σ in Σ̃ which we denote Lk(σ, Σ̃). The underlying poset of cells in Lk(σ, Σ̃) is (W × S^{f})_{>(w,T)}.

Each Coxeter cell is a “simple” polytope. This means that for each pair (σ, τ) where τ is a Coxeter cell and σ is a face of τ, that Lk(σ, τ) is a simplex. It follows that Lk(σ, Σ̃) is a “simplicial cell complex”, in the sense that all its cells are simplices. (We use this term even though it is not a convex cell complex in the strict sense of Remark 2.2.2. We reserve the term “simplicial complex” when the strict property of Remark 2.2.2 is satisfied.)

Example 2.3.1. Let v be the 0-cell in Σ̃_{W} corresponding to the element (1, ∅) in W × S^{f}. We compute Lk(v) (= Lk(v, Σ̃)) in some simple cases.

(i) If W = ℤ/2, then Σ̃_{W} is a circle and Lk(v) = S^{0}.

(ii) If W = (ℤ/2)^{n}, then Σ̃_{W} is a Cartesian product of n circles and Lk(v) is the n-fold join S^{0} ∗ ⋯ ∗ S^{0}. In other words, Lk(v) is the boundary complex of an n-dimensional octahedron.

(iii) Suppose that W is a dihedral group of order 2m and that S = {s_{1}, s_{2}}. Thus, Σ̃_{W} is a 2-complex cellulated by 2m 2m-gons. The complex Lk(v) is 1-dimensional. Its 0-cells correspond to the elements of

(W_{{s_{1}}} × {s_{1}}) ∪ (W_{{s_{2}}} × {s_{2}}) .

Hence, there are four 0-cells. There are 2m edges corresponding to the elements of W. In fact, Lk(v) is as pictured below.

Remark 2.3.2. Suppose that W is the dihedral group of order 2m and that m > 2. Then the link pictured above has cycles consisting of two edges. Thus, there is no way to assign lengths in (0, π) to these edges so that this link is large (that is, so it does not contain cycles of length < 2π). It follows from this that any piecewise Euclidean structure on Σ̃_{W} which is compatible with the given cell structure and with the W -action cannot satisfy CAT (0) (see [G] for the definition of the “ CAT ” inequalities). On the other hand, if m = 2, then Σ̃ is a flat 2-torus.

3. The right-angled case

A Coxeter system (W, S) is right-angled if given any two distinct elements s and s′ of S, the order of ss′ is either 2 or ∞ (i.e., if each off-diagonal entry of the Coxeter matrix is either 2 or ∞).

Artin groups associated to right-angled Coxeter groups are known as graph groups; they are “graph products” of groups where each vertex groups is infinite cyclic. (See [C], [HM].)

We will say that a simplicial cell complex is a flag complex if it is a genuine simplicial complex and if given any set of vertices which are pairwise joined by edges, then this set actually spans a simplex. If W is right-angled, then it follows immediately from the definitions that the geometric realization K_{0} of S_{>∅}^{f} is a flag complex. Hence, as explained in §1.4, the Main Conjecture holds in this case.

3.1. Nonpositive curvature.

If W is finite and right-angled, then W ≅ (ℤ/2)^{n} for some n and the Coxeter cell P_{W} can be taken to be isometric to a regular Euclidean n-cube.

It then follows from §2.2 that if W is an arbitrary right-angled Coxeter group, then both the Salvetti complex Σ̃_{W} and its quotient Z_{W} naturally have the structure of piecewise Euclidean complexes in which each cell is a regular Euclidean cube.

Such a piecewise Euclidean complex is nonpositively curved if, locally, all geodesic triangles satisfy Gromov’s CAT (0)-inequalities (cf. [G, p. 119]). Moreover, if each cell in the complex is a regular cube, then nonpositive curvature is equivalent to the condition that the link of each vertex is a flag complex (cf. [G, p. 122] and [M]).

The main result of this subsection is the following.

Theorem 3.1.1. Suppose W is right-angled. Then Z_{W} with its natural piecewise Euclidean, cubical structure is nonpositively curved.

Remark 3.1.2. Any nonpositively curved, piecewise Euclidean, finite complex is an Eilenberg-MacLane space ([G, p. 119]). Hence, in the case where W is right-angled, the above theorem gives an alternate proof of the main result of [CD1]. The fact that Z_{W} is aspherical is claimed as Theorem 10 in [KR]; the proof given there is incorrect as it is based on a false lemma ([KR, Lemma 9]). However, there is a simple and direct argument for this along the lines indicated in the remark following Lemma 4.3.7 of [CD1].

According to the remarks preceding the theorem, the statement that Z_{W} is nonpositively curved is equivalent to the statement that the link of its 0-cell is a flag complex. Equivalently, we can consider a 0-cell in Σ̃_{W}. Thus, the theorem is an immediate consequence of the following.

Lemma 3.1.3. Suppose W is right-angled and that v is the 0-cell of Σ̃_{W} corresponding to (1, ∅) ∈ W × S^{f}. Then Lk(v, Σ̃_{W}) is a flag complex.

Proof. We first show that Lk(v, Σ̃_{W}) is a simplicial complex. For this, we must show that (W × S^{f})_{>(1,∅)} is isomorphic to the poset of simplices in a simplicial complex. Given (w, T) ∈ (W × S^{f})_{>(1,∅)}, let (W × S^{f})_{((1,∅),(w,T)]} denote the “halfopen interval” between (1, ∅) and (w, T), i.e., it is

{(w′, T′) ∈ W × S^{f} | (1, ∅) < (w′, T′) ≤ (w, T)} .

We must show that (W × S^{f})_{((1,∅),(w,T)]} is isomorphic to the poset of all nonempty subsets of T. Since (w, T) > (1, ∅), w ∈ W_{T}. Let t_{1}, …, t_{n} be the elements of T. Since W_{T} ≅ (ℤ/2)^{n}, any w ∈ W_{T} can be put in the form

(∗)  w = t_{1}^{ε_{1}} … t_{n}^{ε_{n}}

where each ε_{i} is either 0 or 1. Then (w′, T′) < (w, T) if and only if T′ < T and the expression for w′ is obtained from (∗) by deleting those t_{i} which lie in T − T′. In other words, given (w, T) and T′ with (1, ∅) < (w, T), and T′ ≤ T, there is a unique element w′ such that (1, ∅) < (w′, T′) ≤ (w, T). Thus, (W × S^{f})_{>(1,∅)} is an abstract simplicial complex as claimed.

The vertices of Lk(v, Σ̃_{W}) correspond to those elements (w, T) in (W ×S^{f})_{>(1,∅)} such that T is a singleton. Hence, a vertex corresponds to an element (w, {t}) where w = t or w = 1. Suppose {(w_{0}, {t_{0}}), …, (w_{k}, {t_{k}})} corresponds to a set of distinct vertices which are pairwise joined by edges. Put T = {t_{0}, …, t_{k}}. The condition that (w_{i}, {t_{i}}) is joined by an edge to (w_{j}, {t_{j}}) means that t_{i}t_{j} has order 2 and hence, that W_{T} = (ℤ/2)^{k+1}. Thus, T ∈ S^{f}. Let w = w_{1}w_{2} … w_{k} ∈ W_{T}. By the discussion above, (w, T) is a k-simplex of (W × S^{f})_{>(1,∅)} whose vertices are (w_{0}, {t_{0}}), …, (w_{k}, {t_{k}}). Thus, Lk(v, Σ̃_{W}) is a flag complex. □

3.2. Cohomology.

Proposition 3.2.1. Suppose W is right-angled. The CW structure on Z_{W} is “perfect” in the sense of Morse theory. That is to say, in the cellular chain complex, C_{∗}(Z_{W}), all boundary maps are 0.

Proof. As pointed out in [KR, p. 180] the space Z_{W} can be identified with a subcomplex of the torus (S^{1})^{S} with its standard cubical cell structure: Z_{W} = {(x_{s})_{s∈S} ∈ (S^{1})^{S} | if s and t do not commute, then either x_{s} = 1 or x_{t} = 1}. In other words, the i-cell corresponding to T, T ⊂ S and Card (T) = i, belongs to Z_{W} if and only if T ∈ S^{f}. It follows that the cellular chain complex for Z_{W} injects into that of the torus. In the cellular chain complex of a torus, all boundary maps are 0; hence, the same is true in Z_{W}. □

Corollary 3.2.2. Suppose W is right-angled. Then H_{k}(A_{W}) (= H_{k}(Z_{W})) is free abelian. Its rank is the number of elements T in S^{f} such that Card (T) = k.

Remark 3.2.3. The homology of Z_{W} was calculated by Kim and Roush in [KR, Cor. 11].

Still supposing W is right-angled, we turn now to the calculation of the ring structure of H^{∗}(A_{W}). The k-cells of Z_{W} are in one-to-one correspondence with those subsets T of S such that T ∈ S^{f} and Card (T) = k. Order the elements of S, s_{1}, …, s_{n}. Let e_{i} denote the 1-cell corresponding to {s_{i}}. Choose an orientation for e_{i}. If T = {s_{i_{1}}, …, s_{i_{k}}}, with i_{1} < ⋯ < i_{k}, is an element of S^{f}, then the oriented kcell e_{T} corresponding to T is the Cartesian product: e_{T} = e_{i_{1}} ×⋯×e_{i_{k}}. The group of cellular 1-chains C_{1}(Z_{W}) is the free abelian group on {e_{1}, …, e_{n}}. Let z_{1}, …, z_{n} be the dual basis for C^{1}(Z_{W}), the 1-cochains. By Proposition 3.2.1 all boundary maps and coboundary maps are zero; hence we can safely blur the distinctions between cochains, cocycles and cohomology classes. If T = {s_{i_{1}}, …, s_{i_{k}}}, then put

z_{T} = z_{i_{1}} ∪ ⋯ ∪ z_{i_{k}} .

Then z_{T} (e_{T}) = 1 and z_{T} (e_{T′}) = 0 if T′ ≠ T. Thus, {e_{T} } and {z_{T} }, T ∈ S^{f}, Card (T) = k, are dual bases for C_{k}(Z_{W}) and C^{k}(Z_{W}). From the above remarks we can easily deduce the following theorem.

Theorem 3.2.4. Suppose (W, S) is right-angled. Let n = Card (S) and let Λ[y_{1}, …, y_{n}] be the exterior algebra (over ℤ) on indeterminates y_{1}, …, y_{n}. Let I be the ideal generated by all products y_{i}y_{j} such that s_{i}s_{j} has infinite order in W. Then the map y_{i} → z_{i} defines an isomorphism of graded rings φ: Λ[y_{1}, …, y_{n}]/I → H^{∗}(Z_{W}). Thus,

H^{∗}(A_{W}) ≅ Λ[y_{1}, …, y_{n}]/I .

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