THE K(π, 1)-PROBLEM FOR HYPERPLANE COMPLEMENTS

ASSOCIATED TO INFINITE REFLECTION GROUPS

Ruth Charney (*) and Michael W. Davis (*)

Introduction. We begin by recalling some well-known facts. The natural action of the symmetric group S_{n} on ℝ^{n} can be viewed as a group generated by reflections. The reflections in S_{n} are the orthogonal reflections across the hyperplanes H_{ij} = {x ∈ ℝ^{n}|x_{i} = x_{j}}, 1 ≤ i < j ≤ n. The S_{n}-action extends to ℂ^{n} (= ℝ^{n} ⊗ ℂ) and the set of points with nontrivial isotropy group is ⋃ H_{ij} ⊗ ℂ. Let M denote the quotient manifold,

M = (ℂ^{n} − ⋃ H_{ij} ⊗ ℂ)/S_{n}.

Classically, one knows that the fundamental group of M is the braid group B_{n}. In [FN], it is proved that M is an Eilenberg-MacLane space K(B_{n}, 1).

These facts were generalized by Brieskorn [Bn] and Deligne [D] to finite reflection groups as follows. Associated to a Coxeter system (W, S) there is an Artin group A (or “generalized braid group”) constructed by deleting some of the relations in the standard presentation for W. (This is explained in Section 1.) When the Coxeter group W is finite, we say A is of finite type. In this case, there is a (unique) representation of W as an orthogonal reflection group on ℝ^{n} (with n = Card (S)). Consider the induced W -action on ℂ^{n} and the quotient manifold

M = (ℂ^{n} − ⋃ reflection hyperplanes)/W.

In [Bn], Brieskorn proved that the fundamental group of M is A and in [D], Deligne proved that M is an Eilenberg-MacLane space.

As explained below, there is a natural guess as to how this should generalize to infinite Coxeter groups. According to [L] this conjecture is due to Arnold, Pham and Thom. In this paper we shall prove this conjecture for almost all infinite Coxeter groups.

In order to formulate the conjecture we first need to explain the notion of a “linear reflection group” in the case when the group is infinite. A linear reflection on a real vector space V is a linear involution with fixed subspace a hyperplane. (Since there is no

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 51F15, 20F36.

(*) Both authors were partially supported by NSF grants and the Institute for Advanced Study during the preparation of this paper.

assumption of orthogonality, the reflection is not determined by its hyperplane.) Let \overline{C} be a polyhedral cone of full dimension in V and let C denote its interior. Let H_{1}, ⋯, H_{k} be the hyperplanes spanned by the codimension one faces of \overline{C}. For each i, let s_{i} be a linear reflection with fixed hyperplane H_{i} and let W be the subgroup of GL(V) generated by {s_{1}, ⋯, s_{k}}. Then W is called a linear reflection group if w(C) ∩ C = ∅ for all w ∈ W with w ≠ 1.

For each point x in \overline{C} let W_{x} denote the sugroup of W generated by the reflections across the hyperplanes of \overline{C} which contain x and let

C^{f} = \{x ∈ \overline{C} | W_{x} is finite\}.

Given a Coxeter system (W, S), let S^{f} be the poset of those subsets T of S such that the subgroup W_{T} generated by T is finite.

There is the following basic result of [V].

Theorem. (Vinberg). Let W be a linear reflection group on V and let

\overline{I} = ⋃_{w∈W} w\overline{C}.

Then the following statements hold.

(i) (W, {s_{i}}) is a Coxeter system.

(ii) \overline{I} is a convex cone.

(iii) The interior I of \overline{I} is W -stable and W acts properly on I.

(iv) I ∩ \overline{C} = C^{f}

(v) S^{f} can be idenitified with the opposite poset to the poset of faces of C^{f}.

The cone \overline{I} is called the Tits cone.

Tits proved that any Coxeter group W admits a faithful representation as a linear reflection group with \overline{C} a simplicial cone (see [Bo, Ch. V, §4.4 - 4.6]). In general, however, as shown in [V], the cone \overline{C} need not be simplicial and there may be many possible representations of W as a linear reflection group.

Infinite linear reflections groups are important in the theory of Kac-Moody Lie algebras. A central role is played by the domain Ω in V ⊗ ℂ defined by

Ω = {v ∈ V ⊗ ℂ|Im(v) ∈ I}
= V + iI.

By Vinberg’s Theorem, W acts properly on Ω and freely on (Ω−⋃ reflection hyperplanes). Let

M = (Ω − ⋃ reflection hyperplanes)/W.

In various cases, the manifold M occurs in algebraic geometry as the complement of the discriminant of the semiuniversal deformation of a singularity.

In his thesis [L], H. van der Lek proved that the fundamental group of M is the Artin group A associated to W. (A special case had been proved earlier in [N].) The conjecture which we are interested in is the following.

THE K(π, 1)-PROBLEM FOR HYPERPLANE COMPLEMENTS 3

Conjecture 1. (Arnold, Pham, Thom). M is an Eilenberg-MacLane space.

Some progess on this conjecture was made in [L].

Our approach to Conjecture 1 has three steps.

Step 1. There is a combinatorial reformulation of the conjecture which shows that its truth depends only on the Coxeter system (W, S) and not on its particular represenation as a linear reflection group. This is explained in §2.

Step 2. Associated to the Artin group A there is a cetain simplicial complex Φ, which we define in §1 and call the “modified Deligne complex”. Using the results of [D], it can be shown that Φ is simply connected and that Conjecture 1 is equivalent to the following.

Conjecture 2. Φ is contractible.

Step 3. The complex Φ can be given a piecewise Euclidean structure. When this structure is CAT (0), in the sense of Gromov [G], then the contractiblity of Φ follows. Since Φ is simply connected, this reduces to checking that the induced piecewise spherical metric on the link of each vertex is CAT (1).

Step 1 is accomplished in §2 and Step 2 in §1 and §3. The core of the paper lies in the work on Step 3 in §4.

Associated to any Coxeter system (W, S) is a “Coxeter complex” Σ̂ on which W acts with “fundamental chamber” a simplex of dimension n = Card (S)−1. If W is finite, then Σ̂ can be identified with the sphere 𝕊^{n}, and we call Σ̂ spherical. An analogous complex Φ̂, called the “Deligne complex” (since it was introduced in [D]) can be constructed for the associated Artin group A. The Deligne complex is “building-like” in the sense that it is made up of subcomplexes (“apartments”) each isomorphic to Σ̂. If A is of finite type, we say that Φ̂ is spherical.

In light of our definition of Ω in terms of the Tits cone, part (iv) of Vinberg’s Theorem indicates that we need a complex in which only finite subgroups of W appear as isotropy groups. We therefore consider a “modified Coxeter complex” Σ with isotropy groups W_{T} , T ∈ S^{f}, and “fundamental chamber” K, which is the geometric realization of the derived complex of S^{f}. The complex Φ which arises in Step 2 is the analoguous complex for A, with fundamental chamber K and with isotropy groups the sub-Artin groups A_{T} of finite type. Φ is “building-like” with “apartments” isomorphic to Σ. Although the proof of Step 2 was inspired by the arguments in [D] and [L], these arguments are greatly simplified by using the theory of complexes of groups developed by Haefliger in [H1] and [H2]. For an arbitrary Coxeter system (W, S) with associated Artin group A, we show that the subgroups A_{T} of A, generated by T ∈ S^{f}, form a complex of groups. This complex of groups is developable, that is, it arises from the action of a group on a simplicial complex, explicitly, from the action of A on the modified Deligne complex Φ. We prove that the quotient manifold M is homotopy equivalent to the classifying space of this complex of groups and that Φ is homotopy equivalent to the universal covering space of this classifying space. The equivalence of Conjectures 1 and 2 follows.

We turn now to Step 3. Our program for proving the contractibility of Φ was motivated by the proofs of Gromov [G] and Moussong [M] that Σ can be given a piecewise Euclidean CAT (0) metric. The complex Σ may be viewed as a quotient of W × K, with copies of K glued along certain faces. K, and hence Σ, has a natural decomposition into combinatorial cubes. If we assign each of these cubes the metric of a regular Euclidean cube (as Gromov suggests), then the condition that links of vertices are CAT (1) reduces to a combinatorial condition, namely that the links are “flag complexes” (see §4.2). Analyzing the links of vertices in Σ, we find that the links are joins of two types of complexes:

(i) the spherical Coxeter complex Σ̂_{T} associated to some finite W_{T},

(ii) the geometric realization θ_{T} of the abstract simplicial complex

S_{>T}^{f} = {T′ ∈ S^{f}|T ⫋ T′},

where T ∈ S^{f}. It is well-known (cf., [Br, p. 29]) that the first of these is always a flag complex. The second is a flag complex for all T ∈ S^{f} if and only if (W, S) satisfies the following condition.

(FC): If T ⊂ S and every pair of elements in T generates a finite subgroup of W, then T generates a finite subgroup of W.

Thus, Gromov’s piecewise Euclidean structure on Σ (using regular cubes) is CAT (0) if and only if (W, S) satisfies (F C).

The situation for Φ is similar, Φ may be viewed as a quotient of A × K and, as such, has a decomposition into combinatorial cubes which we can view as regular Euclidean cubes. In this case, the link of a vertex is a join of a spherical Deligne complex Φ̂_{T} and the complex θ_{T}. We prove that every spherical Deligne complex is a flag complex. We conclude that, as for Σ, this metric on Φ is CAT (0) if and only if (W, S) satisfies (F C). This gives our main theorem.

Theorem A. Conjecture 1 holds for any Coxeter system (W, S) satisfying (F C).

If W_{T} is a minimal example for which (F C) fails, then its fundamental chamber must be a simplex of dimension at least 2. It is well-known (cf., [Bo] or [La] that such a Coxeter group is either an irreducible Euclidean reflection group (= an “affine Weyl group”) or a hyperbolic reflection group. Moreover, in the hyperbolic case the dimension must be ≤ 4. (In dimension 2 these are the hyperbolic triangle groups, there are 9 more examples in dimension 4 and 5 more in dimension 4.) Thus, condition (F C) holds for (W, S) whenever its Coxeter diagram does not contain a subdiagram of one of the above types. These forbidden subdiagrams are listed in Tables 1 and 2 in §4.3.

In his Ph.D. thesis [M], G. Moussong considered a different piecewise Euclidean metric on Σ. He identifies each combinatorial cube in K with a certain Euclidean convex polytope, the dihedral angles of which depend on W. These angles are defined in such a way that the induced metric on each spherical Coxeter complex Σ̂_{T} appearing in the link of a vertex is the natural metric, that is, the standard round metric on the sphere. Moussong proves that the induced piecewise spherical metrics on the complexes θ_{T} appearing in the links are CAT (1) for all T. Since joins of CAT (1) spaces are CAT (1), it follows that Moussong’s metric on Σ is CAT (0) for all (W, S).

If we put the analoguous metric on Φ, the complexes θ_{T} in the links are again CAT (1) by Moussong’s result, and every apartment in Φ̂_{T} is a round sphere, but it does not follow a priori that Φ̂_{T} is CAT (1). Hence, Moussong’s metric on Φ is CAT (0) if and only if the following conjecture holds.

Conjecture 3. For each finite Coxeter system the natural metric on the associated Deligne complex Φ̂ (making each apartment into a round sphere) is CAT (1).

Thus, Conjecture 3 implies Conjecture 1. There is one case when Conjecture 3 is easy to prove: when the Coxeter group is a finite dihedral group. In this case Conjecture 3 follows from a lemma of Appel and Schupp [AS]. This is enough to prove Conjecture 1 whenever K is two-dimensional; that is, when any three distinct elements of S generate an infinite subgroup of W. This gives the following.

Theorem B. Conjecture 1 holds for any Coxeter system such that K is two-dimensional.

For example, Theorem B applies to Euclidean and hyperbolic triangle groups. It also applies in the case when, in the language of [AS], (W, S) is of “large type”. In this case, Theorem B was proved by Hendriks [He].

In a subsequent paper, [CD3], we describe a finite CW -complex which is homotopy equivalent to M. This complex has the same dimension as K. In the situations where Conjecture 1 holds, this allows us to deduce quite a bit about the Artin group A. For example, its cohomological dimension is equal to dimK.

§1. The Complex of Artin Groups.

In this section we establish some preliminaries on complexes of groups in general and complexes of Coxeter groups and Artin groups in particular.

(1.1) Coxeter groups and Artin groups. A Coxeter matrix is a symmetric matrix M = (m_{ij}) with m_{ii} = 1 for all i and m_{ij} ∈ {2, 3, ⋯, ∞} for i ≠ j. Associated to an n × n Coxeter matrix is a Coxeter system (W, S) where S = {s_{1}, ⋯, s_{n}} is a finite set and W is the group with presentation,

W = ⟨s_{1}, ⋯, s_{n}|(s_{i}s_{j})^{m_{ij}} = 1⟩,

(with the convention that the relation is ignored if m_{ij} = ∞). If T ⊂ S, then let (W_{T} , T) be the Coxeter system defined by the corresponding T × T submatrix of M and let \overline{W_{T}} be the subgroup of W generated by T.

Lemma 1.1.1. ([Bo, Ch IV, §1.8, Théoréme 2 (i)]). The natural map W_{T} → \overline{W_{T}} is an isomorphism.

Hence, we can view W_{T} as a subgroup of W. The groups W_{T} are called the special subgroups of W and their cosets wW_{T} are the special cosets. (By convention, W_{∅} = {1} and W_{∅} is also a special subgroup.)

Also associated to the Coxeter matrix (m_{ij}) is an Artin group (or “generalized braid group”)

A = ⟨s_{1}, ⋯, s_{n}|prod(s_{i}, s_{j}; m_{ij}) = prod(s_{j}, s_{i}; m_{ij})⟩

where

prod(s_{i}, s_{j}; m_{ij}) = \underbrace{s_{i}s_{j}s_{i}s_{j} ⋯}_{m_{ij}−terms}

If the corresponding Coxeter group W is finite, we say A is of finite type. If we add to the presentation for A the relations s_{i}^{2} = 1, then we get the presentation for W; hence, W is a quotient of A. Denote the quotient homomorphism by q : A → W.

For T ⊂ S, let A_{T} denote the Artin group associated to (W_{T} , T) and let \overline{A_{T}} be the subgroup of A generated by T.

Lemma 1.1.2. The natural map A_{T} → \overline{A_{T}} is an isomorphism.

This was proved by Deligne [D, Théoremè 4.14(iii)] for A of finite type and by van der Lek [L, Theorem 4.13] for general A. We include our own proof in the case that A_{T} is of finite type in §3, as Corollary 3.25. Hence, we can identify A_{T} with its image in A. These subgroups are called the special subgroups of A.

The quotient homomorphism q : A → W has a set theoretic section m : W → A defined as follows. For w ∈ W, let s_{1} ⋯ s_{k}, s_{i} ∈ S, be a word of minimal length representing w in the free group F (S). Define m(w) to be the image of s_{1} ⋯ s_{k} in A. It follows from Tit’s solution to the word problem for Coxeter groups in [T] that m is well-defined. The mapping m is not, of course, a homorphism, but it does preserve special cosets.

Lemma 1.1.3. If w_{1}W_{T} = w_{2}W_{T}, then m(w_{1})A_{T} = m(w_{2})A_{T}.

Proof. It follows from [Bo, Ch IV , Ex. 3, p. 37] that a special coset wW_{T} has a unique element w_{0} of minimal length and that for any w ∈ w_{0}W_{T}, we can write w = w_{0}u with u ∈ W_{T} and length (w) = length (w_{0})+ length (u). It then follows from the definition of m that m(w) = m(w_{0})m(u). A minimal word representing u will involve only the generators in T [Bo, Ch IV, §1.8], hence m(u) ∈ A_{T} and m(w)A_{T} = m(w_{0})A_{T} □

Given a Coxeter system (W, S), we define the Coxeter complex Σ̂ as follows. For n = Card (S) − 1, let Δ_{S} be an n-simplex with vertices labeled by the elements of S. Denote by σ_{T} the face of Δ_{S} spanned by the vertices {s_{i}|s_{i} ∉ T }. Define

Σ̂ = W × Δ_{S}/ ∼

where (w_{1}, x_{1}) ∼ (w_{2}, x_{2}) providing x_{1} = x_{2} and if σ_{T} is the (open) simplex containing x_{1}, then w_{1}^{−1}w_{2} ∈ W_{T}.

There is an analogous complex for A, first introduced in [D, §2] which we call the Deligne complex and denote by Φ̂. Namely

Φ̂ = A × Δ_{S}/ ∼

where (a_{1}, x_{1}) ∼ (a_{2}, x_{2}) providing x_{1} = x_{2} and if σ_{T} is the (open) simplex containing x_{1}, then a_{1}^{−1}a_{2} ∈ A_{T}.

The natural map Δ_{S} = 1 × Δ_{S} → Σ̂ is an injection. Its image is called the fundamental chamber of Σ̂; its translates under W are chambers of Σ̂. Likewise for Δ_{S} = 1 × Δ_{S} → Φ̂.

The quotient map q : A → W induces a simplicial map q̂ : Φ̂ → Σ̂. It follows from Lemma 1.1.3 that the (set-theoretic) section m : W → A of q induces a simplicial section m̂ : Σ̂ → Φ̂ of q̂. The image of m̂ in Φ̂ is called the fundamental apartment of Φ̂; its translates under A are called apartments. Since q̂ is equivariant (with respect to q), it maps each apartment isomorphically onto Σ̂.

Remark. The terminology of “chambers” and “apartments” is borrowed from the theory of buildings. In some respects Φ̂ is “building-like”, but it fails to satisfy the axiom for buildings which requires that any two chambers belong to a common apartment.

(1.2) Posets. Let P be a partially ordered set (or poset). Associated to P is the derived poset P′ whose elements are totally ordered chains σ = (a_{0} < a_{1} < ⋯ < a_{k}) in P with τ < σ if τ is a subchain of σ. The derived poset may be viewed as an abstract simplicial complex. We denote its geometric realization by |P′|.

For an element a ∈ P, define subposets

P_{<a} = {b ∈ P|b < a}

and similarly, P_{≤a}, P_{>a} and P_{≥a}. Let K = |P′|. The subcomplex

F_{a} = |(P_{≥a})′|

of K is called the face associated to a. The open face associated to a is the relative interior of F_{a},

F^{∘}_{a} = |(P_{≥a})′| − |(P_{>a})′|.

The open faces partition K; that is, every point of K belongs to a unique open face. Similarly, we define the dual face and the open dual face associated to a by

F_{a}^{∗} = |(P_{≤a})′|
F^{∘} _{a}^{∗} = |(P_{≤a})′| − |(P_{<a})′|

Note that if P^{op} denotes the poset obtained by reversing the ordering on P, then P′ and (P^{op})′ are isomorphic posets and the faces of |P′| are the dual faces of |(P^{op})′| and vice versa.

Example. Let X be a cell complex and let P be the set of cells of X ordered by reverse inclusion. Then |P′| can be naturally identified with the barycentric subdivision of X.

The face F_{c} associated to a cell c ⊂ X is then identified with c, and the dual face F_{c}^{∗} is identified with the cone on the link of c in X.

(1.3) Complexes of groups. The notion of a “complex of groups” was introduced by A. Haefliger in [H1], [H2]. (Also, see [Co] and [S].) We review some basic material here and refer the reader to [H1], [H2] for more details.

Definition. Suppose P is a poset and X is the geometric realization of P′. A simple complex of groups G(X) on X is given by the following data:

(1) for each a ∈ P, a group G_{a},

(2) for each a, b ∈ P with b < a an injective homomorphism ψ_{ab} : G_{b} → G_{a} such that if c < b < a, then ψ_{ac} = ψ_{ab} ∘ ψ_{bc}.

Remark. Haefliger’s definition of a “complex of groups” in [H2] is more general than the above in two ways. First of all, in place of the poset P, he allows a small category without loop. Secondly, the homomorphism ψ_{ac} is required to agree with the composition ψ_{ab} ∘ ψ_{bc} only up to conjugation by an element of G_{a} (and this element must be kept track of). This more general definition plays no role in our situation, hence the term simple complex of groups.

Associated to a complex of groups G(X) there is a classifying space BG(X). In the case of a simple complex of groups it can be defined as follows. Let CG(X) be the category with objects the elements of P and with

Hom(b, a) = \{ \begin{matrix}G_{a} & , if b ≤ a \\ ∅ & , otherwise. \end{matrix}

For c < b < a, the composition of g ∈ Hom(c, b) with h ∈ Hom(b, a) is defined to be hψ_{ab}(g) ∈ G_{a} = Hom(c, a). The classifying space BG(X) is the geometric realization of the nerve of the category CG(X). There is an obvious functor from CG(X) to P (viewed as a category) which gives rise to a projection p : BG(X) → X.

The fundamental group, π_{1}(G(X)), is the fundamental group of the space BG(X). An alternate description of π_{1}(G(X)) is given in [H1, §3] with generators and relations

being given in terms of the groups G_{a} and the edges of X. From this description, one can immediately deduce the following.

Lemma 1.3.1. If G(X) is a simple complex of groups and X is simply connected, then π_{1}(G(X)) is the colimit of the groups G_{a} with respect to the homomorphisms ψ_{ab}. In other words, if ⟨S_{a}|R_{a}⟩ is a presentation for G_{a}, then

⟨⋃_{a} S_{a}| ⋃_{a} R_{a}, ψ_{ab}(s) = s, ∀b < a, s ∈ S_{b}⟩

is a presentation for π_{1}(G(X)).

Suppose P̃ is a poset and X̃ is the geometric realization of its derived complex. (Then X̃ is an “ordered” simplicial complex in the sense that the vertex set of each of its simplices is canonically ordered.) An order-preserving G-action on P̃ is without inversions if given ã, b̃ ∈ P̃, with b̃ < ã, then the isotropy subgroups Gb̃, Gã of b̃ and ã satisfy Gb̃ ⊂ Gã.

Suppose now that G acts without inversions on P̃. There is an induced simplicial Gaction on X̃. Set P = P̃/G and X = X̃/G. Then P is a “small category without loop”. For simplicity, assume that P is a poset, i.e. there is at most one morphism between any two objects of P. (This holds if we strengthen the “action without inversion” condition to read: if ã < b̃ and g · ã < b̃, then g ∈ Gb̃.) Then X is the geometric realization of P′, and we can associate a complex of groups G(X) to this situation as follows. Choose a representative ã for each G-orbit a ∈ P. The group associated to a is the isotropy subgroup Gã. For the moment fix b ∈ P. If b < a, then, since the action is without inversions, there is a unique element ã′ in the orbit a, such that b̃ < ã′. Hence, there is an element g_{ab} ∈ G such that g_{ab}ã′ = ã. Define ψ_{ab} : Gb̃ → Gã to be the composition

Gb̃ ⊂ G_{}_{ã′} ⟶^{Ad(g_{ab})} Gã

This gives the data for a complex of groups G(X). (The definition in [H2, p. 281] is designed so that this will be true.) A complex of groups arising in this fashion is called developable.

In general, of course, G(X) will not be simple. However, if P has a smallest element (as will be the case in our application), then the choices above can be modified so that each g_{ab} = 1, i.e., so that G(X) is simple (cf., Proposition 2.2.1 in [H2]).

Theorem 1.3.2. (Haefliger [H1]). A complex of groups G(X) is developable if and only if the natural maps G_{a} → π_{1}(G(X)) are injective for all a ∈ P.

Suppose that G, P̃, X̃ and G(X) are as above. Then the classifying space of G(X) can be identified with the Borel construction

BG(X) = X̃ ×_{G} EG

where EG is the universal covering space of BG(= K(G, 1)) cf., [H2, Proposition 3.2.3]. It follows that there is a short exact sequence

1 → π_{1}(X̃) → π_{1}(G(X)) → G → 1.

Thus, X̃ is simply connected if and only if π_{1}(G(X)) = G. Moreover, if this is the case then the universal cover of BG(X) is homotopy equivalent to X̃.

(1.4) Complexes of spaces. In [H2], Haefliger also introduces the notion of a complex of spaces on an ordered simplicial cell complex X. We again restrict our attention to the case in which X = |P′| for a poset P.

Definition. A simple complex of spaces over X is a topological space Y together with a continuous projection p : Y → X and a section s : X^{(1)} → Y of p over the 1-skeleton of X satisfying the following conditions.

(i) For each vertex a of X, p^{−1}(a) is connected and there is a retraction r_{a} : p^{−1}(F_{a}^{∗}) → p^{−1}(a) which is homotopic to the identity relative to p^{−1}(a). Furthermore, r_{a} ∘ s must be constant on X^{(1)} ∩ F_{a}^{∗}.

(ii) Let G_{a} = π_{1}(p^{−1}(a), s(a)). For b < a, let ψ_{ab} : G_{b} → G_{a} be the map of fundamental groups induced by the composition p^{−1}(b) ↪ p^{−1}(F_{a}^{∗}) ⟶^{r_{a}} p^{−1}(a). Then ψ_{ab} is injective.

The data {G_{a}, ψ_{ab}} gives rise to a simple complex of groups. (In the general definition of a “complex of spaces” in [H2], as before, one relaxes the condition on P to be a small category without loop and in (i) it is only required that r_{a} ∘ s be constant on the vertices of F_{a}^{∗}.)

Definition. If p : Y → X is a complex of spaces with associated complex of groups G(X), we say that Y is a topological realization of G(X). If Y is a CW -complex and p is cellular, then Y is a cellular realization of G(X). If p^{−1}(a) is aspherical for every vertex a, then Y is an aspherical realization of G(X).

An example of an aspherical cellular realization of G(X) is the classifying space BG(X).

Theorem 1.4.1. ([H2, Theorem 3.5.2]). Every aspherical cellular realization of G(X) is homotopy equivalent to BG(X).

Remark. It follows from van Kampen’s Theorem that if Y is a cellular realization of G(X), then π_{1}(Y) = π_{1}(G(X)) (cf. [Co], Theorem 2.4).

(1.5) Complexes of Coxeter groups and Artin groups. Suppose (W, S) is a Coxeter system. Let S be the poset of subsets of S ordered by inclusion, let S^{f} be the suposet defined by

S^{f} = {T ⊆ S | W_{T} is finite}.

and let K = |(S^{f})′|. Let G stand for W or A, where A is the Artin group associated to (W, S). It follows from Lemmas 1.1.1 and 1.1.2, that the groups G_{T} and natural inclusions G_{T} → G_{T′} for T < T′ define a simple complex of groups G(K) and that this complex of groups is developable. Since S^{f} has an initial object, ∅, K is simply connected, so it follows from Lemma 1.3.1 that the π_{1}(G(X)) = G.

Remarks. (i) We could have considered the analogous complex of groups over the full poset S. In later sections, however, it will be important that the vertex groups G_{T} are all finite (for G = W) or of finite type (for G = A).

(ii) In his Ph.D. thesis [S], B. Spieler showed that any complex of groups which supports a metric of “nonpositive curvature” (i.e., is locally CAT (0)) is developable. Moussong [M] showed that W (K) always supports such a metric. In Section 4 we discuss the problem of assigning nonpositively curved metrics to A(K).

Since G(K) is developable, it arises from the action of G on a poset P̃. The geometric realizations of the posets P̃ for G = W and G = A will play a major role in the sections which follow so we now describe them explicitly.

For G = W or A, let

GS^{f} = {gG_{T} |g ∈ G, T ∈ S^{f}}

ordered by inclusion. There is an obvious order-preserving left action of G on GS^{f}. The stabilizer of gG_{T} ∈ GS^{f} is the subgroup gG_{T} g^{−1}; hence, G acts without inversion on GS^{f}.

The inclusion of S^{f} into GS^{f} as the cosets eG_{T} of the identity element e gives a splitting of the projection GS^{f} → GS^{f}/G = S^{f}. Using this splitting to choose representives for S^{f} in GS^{f}, the complex of groups associated to the action of G on GS^{f} is precisely the complex G(K) described above.

Let K̃_{G} be the geometric realization of the derived complex of GS^{f}. We have already observed that π_{1}(G(K)) = G. Thus, BG(K) = K̃_{G} ×_{G} EG and its universal covering space is given by B̃G(K) = K̃_{G} × EG, which is homotopy equivalent to K̃_{G}.

Henceforth, we will use the following notation,

Σ = K̃_{W} = |(W S^{f})′|
Φ = K̃_{A} = |(AS^{f})′|

The complex Σ will be called the modified Coxeter complex and Φ the modified Deligne complex. The discussion above is summarized in the theorem below.

Theorem 1.5.1.

(i) The complex of Coxeter groups W (K) is developable and its fundamental group is W. Its classifying space BW (K) is

Σ ×_{W} EW

and hence, the universal covering space of BW (K) is homotopy equivalent to the modified Coxeter complex Σ.

(ii) The complex of Artin groups A(K) is developable and its fundamental group is A. Its classifying space BA(K) is

Φ ×_{A} EA

and hence, the universal covering space of BA(K) is homotopy equivalent to the modified Deligne complex Φ.

As suggested by the terminology, the complexes Σ and Φ are closely related to the ordinary Coxeter and Deligne complexes Σ̂ and Φ̂, at least in the case when W is finite. To see this, note that the inclusion S^{f} ↪ W S^{f} corresponds to an inclusion K ↪ Σ. The W -translates of K cover Σ, so Σ is a quotient of W × K, namely

Σ = W × K/ ∼

where (w_{1}, x_{1}) ∼ (w_{2}, x_{2}) providing x_{1} = x_{2} in K and if F^{∘}_{T} is the open face containing x_{1}, then w_{1}^{−1}w_{2}A ∈ W_{T}. Similarly, we have K ↪ Φ and

Φ = A × K/ ∼

where (a_{1}, x_{1}) ∼ (a_{2}, x_{2}) providing x_{1} = x_{2} in K and if F^{∘}_{T} is the open face containing x_{1}, then a_{1}^{−1}a_{2} ∈ A_{T}. The translates of K in Σ (resp. Φ) are called chambers and K itself is called the fundamental chamber of Σ (resp. Φ).

Now recall the simplex Δ_{S}, with vertices labelled by S, used to define the Coxeter complex Σ̂. With notation as in (1.1), the map σ_{T} ↦ T is an order reversing map from the poset Simp (Δ_{S}) of simplices of Δ_{S} to S. In fact, it is an isomorphism from Simp (Δ_{S}) onto (S_{<S})^{op}. It follows that |S′ | (= |(S^{op})′|)) is naturally isomorphic to the cone on the barycentric subdivision of Δ_{S} where the cone point is the vertex v_{S} corresponding to S ∈ S. For T ≠ S the face F_{T} of |S′| is identified under this isomorphism with the barycentric subdivision of the face σ_{T} of Δ_{S}. Now suppose that W is finite. Then S = S^{f}, so K = |S′|. Thus, K is the cone on the barycentric subdivision of Δ_{S}. Moreover, since the translates of v_{S}(= F_{S}) in Σ are all identified, we conclude that for W finite, Σ is the cone on the barycentric subdivision of Σ̂. Similarly, Φ is the cone on the barycentric subdivision of Φ̂.

As is the case of Σ̂, Φ̂, the projection q : A → W induces a simplicial map \overline{q} : Φ → Σ and the section m : W → A induces a simplicial section \overline{m} : Σ → Φ of \overline{q} (Lemma 1.1.3). The image of \overline{m} in Φ is called the fundamental apartment of Φ and its translates by A are called apartments. Since \overline{q} is equivariant (with respect to q), it maps each apartment isomorphically onto Σ.

§2. Hyperplane complements.

As in the Introduction, suppose (W, S) is a Coxeter system acting as a linear reflection group on a real vector space V and that \overline{I} is the Tits cone. We identify V ⊗ ℂ with V × V and the domain Ω with V × I.

Let R denote the set of elements of W which act as reflections on V. (This is precisely the set of conjugates of the elements in S.) For r ∈ R, let H_{r} denote the hyperplane of V fixed by r. We are interested in the hyperplane complement

Y = (V × I) − ⋃_{r∈R}(H_{r} × H_{r}).

W acts freely on Y and we denote the quotient manifold by M = Y/W.

The goal of this section is to replace Y by a combinatorial object. For r ∈ R, let P_{r} denote the fixed set of r acting on the modified Coxeter complex Σ. P_{r} is called the wall associated to r. Consider the diagonal action of W on Σ × Σ. The set of points with nontrivial isotropy groups is the union of the P_{r} × P_{r} for all walls P_{r} of Σ. Let

Y = (Σ × Σ) − ⋃_{r∈R}(P_{r} × P_{r})

and let Q = Y/W be the orbit space. In this section we prove that Y is W -equivariantly homotopy equivalent to Y and hence, that Q is homotopy equivalent to M.

(2.1) Reduction to I × I. For a group G and a G-space Z, denote the isotropy group at z ∈ Z by G_{z}, (G_{z} = {g ∈ G|gz = z}.) An equivariant map f : Z → Z′ is isovariant if for all z ∈ Z, G_{z} = G_{f(z)}. The map f is an isovariant (resp. equivariant) homotopy equivalence if there is an isovariant (resp. equivariant) map h : Z′ → Z and isovariant (resp. equivariant) homotopies f ∘ h ∼ id_{Z′} and h ∘ f ∼ id_{Z}.

We first reduce the study of the W -space V × I to that of I × I, which can be more easily compared with Σ × Σ.

Lemma 2.1.1. The inclusion i : I × I → V × I is a W -isovariant homotopy equivalence.

Proof. Define equivariant maps p′ and p from V × I to I as follows. If x ∈ I, then set p′(x, y) = x. If x ∉ I, then let p′(x, y) be the point where the line segment from x to y intersects ∂\overline{I}.

Let p(x, y) = \frac{1}{2}p′(x, y) + \frac{1}{2}y. Then p(x, y) is on the line segment from x to y and lies in I.

Define r_{1} : V ×I → I×I by r_{1}(x, y) = (p(x, y), y). Then r_{1} is an equivariant deformation retraction. Indeed, the straight line homotopy

r_{t}(x, y) = (tp(x, y) + (1 − t)x, y)

is the required deformation of i ∘ r_{1} to id_{V ×I}.

It remains to show that r_{t} is isovariant. Note that if w ∈ W fixes two distinct points in V, then it fixes the affine line which they span. Suppose w ∈ W fixes r_{t}(x, y). Then it fixes y and the point tp(x, y) + (1 − t)x on the line from x to y; hence, it also fixes x. Thus, w fixes (x, y), which shows that r_{t} is isovariant. □

(2.2) Mirror structures. A mirror structure on a space X indexed by a set S, is a collection of closed subspaces {X_{s}}_{s∈S}. Given a mirror structure and a subset T of S, define

X_{T} = ⋂_{s∈T} X_{s}

if T is nonempty, and

X_{∅} = X.

The subspaces X_{T} are the faces of X. For x ∈ X, set

S(x) = {s ∈ S|x ∈ X_{s}},

and define

X^{∘}_{T} = {x ∈ X_{T} |S(x) = T }
∂X_{T} = X_{T} − X^{∘}_{T}

If X has a mirror structure indexed by S and T ⊆ S, then it has an induced mirror structure indexed by T (by forgetting the mirrors indexed by S − T).

Suppose that X′ has a mirror structure indexed by S and X has a mirror structure indexed by T, for T ⊆ S. A map f : X → X′ is T -face-preserving if T (x) = T (f(x)) for all x ∈ X. If T = S, then we shall simply say that f is face-preserving.

A mirror structure is collared if for every subset T of S, there exists a map c_{T} : ∂X_{T} × [0, 1] → X_{T} such that c_{T} is a homeomorphism onto a closed neighborhood of ∂X_{T} in X_{T} and such that the restriction of c_{T} to ∂X_{T} × 0 is the inclusion.

Let T be a collection of subsets of S. A mirror structure is defined over T if X_{T} = ∅ for

T ∉ T . It is contractible over T if it is defined over T and if X^{∘}_{T} is contractible for T ∈ T .

Example 2.2.1.

(i) In the notation of §1.5, let X = K(= |(S^{f})′|) and define a mirror structure by X_{s} = F_{{s}}, where F_{T} denotes the face |(S_{≥T}^{f})′| as in §1.3. This mirror structure is defined over S^{f} and X_{T} = F_{T}, for T ∈ S^{f}. Since X_{T} is a cone over ∂X_{T}, it is collared and contractible over S^{f}.

(ii) In the notation of Vinberg’s Theorem (stated in the Introduction), let X = C^{f} and

consider the mirror structure given by X_{s} = C^{f} ∩ H_{s}, which is defined over S^{f}. Since X^{∘}_{T} is a convex subset of V, the mirror structure is contractible over S^{f}. Since X_{T} is a smooth manifold with corners and its bounday is ∂X_{T}, it follows from the Collared Neighborhood Theorem that this mirror structure is collared.

A face-preserving map f : X → X′ is a face-preserving homotopy equivalence if there is a face-preserving map g : X′ → X such that both composites are homotopic to the identity maps via face-preserving homotopies.

Lemma 2.2.2. Let X, X′ be spaces with mirror structures indexed by S and let T be a collection of subsets of S.

(i) Suppose the mirror structure of X is collared and defined over T and suppose the mirror structure on X′ is contractible over T . Then there is a face-preserving map f : X → X′ and it is unique up to face-preserving homotopy.

(ii) If the mirror structures on X and X′ are both collared and contractible over T , then f is a face-preserving homotopy equivalence.

Proof. The proof of (i) is a standard inductive argument. We may suppose by induction

that f is defined on ∂X_{T}. Using the contractibility of X^{∘}′_{T}, we then extend it to a map from X_{T} to X′_{T} which is constant on X_{T} − c_{T} (∂X_{T} × [0, 1]). Similarly, for a homototpy between two such maps. This proves (i). Statement (ii) follows easily from (i). □

Next we generalize the construction of Σ̂ in §1.1. Given a Coxeter system (W, S) and a mirror structure {X_{s}}_{s∈S}; define a space

X (W, X) = (W × X)/ ∼

where (w_{1}, x_{1}) ∼ (w_{2}, x_{2}) if x_{1} = x_{2} and w_{1}^{−1}w_{2} ∈ W_{S(x_{1})}. For example, Σ = X (W, K).

Let [w, x] denote the image of (w, x) in X (W, X). The group W acts on X (W, X) via the formula: w′ · [w, x] = [w′w, x]. In the next lemma we collect some easy facts about this construction. (For the proofs of (i), (ii) and (iii), see Propositions 1, 2 and 4 of [V, pp. 1088-1089].)

Lemma 2.2.3. (Vinberg).

(i) The map X (W, X) → X defined by [w, x] → x induces a homeomorphism X (W, X)/W → X.

(ii) The map X → X (W, X) defined by x → [1, x] is a homeomorphism onto its image (which we identify with X).

(iii) If X is Hausdorff and the mirror structure is defined over S^{f}, then the W -action is proper.

(iv) If Z is a W -space and φ : X → Z is a map such that s(φ(x)) = φ(x) for all s ∈ S and x ∈ X_{s}, then φ extends to a W -equivariant map X (W, X) → Z by the formula: [w, x] → wφ(x).

(v) If f : X → X′ is a face-preserving map of spaces with mirror structures, then f extends to an isovariant map f̂ : X (W, X) → X (W, X′) by [w, x] → [w, f(x)]. If f is a face-preserving homotopy eqivalence, then f̂ is an isovariant homotopy equivalence.

Proposition 2.2.4. There is an isovariant homotopy equivalence between Σ and I.

Proof. Applying part (iv) of Lemma 2.2.3 to the inclusion C^{f} ↪ I we see that X (W, C^{f}) can be identified with I. We also have Σ = X (W, K). As observed in Example 2.2.1, the mirror structures on K and C^{f} are collared and contractible over S^{f}; hence, the proposition follows from Lemma 2.2.2. □

Corollary 2.2.5. Σ × Σ is W -isovariantly homotopy equivalent to I × I and hence, Y is W -equivariantly homotopy equivalent to the hyperplane complement Y.

Suppose X is a space with mirror structure. For each s ∈ S, let H_{{s}}(X) be the half-space of X (W, X) containing X defined by

H_{{s}}(X) = ⋃_{ℓ(ws)>ℓ(w)} wX

where the union is over all w in W with ℓ(ws) > ℓ(w) (ℓ denotes the word length). For T ⊆ S, define

H_{T} (X) = ⋂_{s∈T} H_{{s}}(X).

Then H_{T} (X) has a mirror structure indexed by T defined by intersecting with those walls of X (W, X) which are indexed by T. Applying Lemma 2.2.3 (iv) with W = W_{T} , Z = X (W, X) and φ : H_{T} (X) ↪ X (W, X) the inclusion, we obtain a homeomorphism X (W_{T} , H_{T} (X)) → X (W, X) which we regard as an identification.

For T ∈ S^{f}, let K_{T} and Σ_{T} denote, respectively, the fundamental chamber and modified Coxeter complex for the finite Coxeter system (W_{T} , T). There is a natural identification of K_{T} with the subset F_{T}^{∗} ⊂ K, where F_{T}^{∗} is the dual face of T, discussed in §1.3.

Lemma 2.2.6. The inclusion F_{T}^{∗} ↪ K induces a W_{T} -isovariant homotopy equivalence Σ_{T} ↪ Σ.

Proof. We have,

Σ_{T} = X (W_{T} , F_{T}^{∗}),
Σ = X (W_{T} , H_{T} (K)).

The mirror structures on F_{T}^{∗} and H_{T} (K) are both collared (clearly). By Lemma 2.2.2(ii) and Lemma 2.2.3 (v), it suffices to show these mirror structures are contractible over T , the collection of all subsets of T. This is true for F_{T}^{∗} by Example 2.2.1 (i). To prove the mirror structure on H_{T} (K) is contractible we use the homotopy equivalence Σ → I of Proposition 2.2.4. This restricts to a T -face-preserving homotopy equivalence H_{T} (K) → H_{T} (C^{f}), where H_{T} (C^{f}) is an intersection of half-spaces of I. Since the faces of H_{T} (C^{f}) are defined by intersecting with hyperplanes, they are convex and hence, contractible. □

§3. A complex of spaces.

The projection of Σ × Σ onto the first factor restricts to a W -equivariant map Y → Σ and this induces a map of orbit spaces

p : Q → K

(where Q = Y/W and K = Σ/W). We set up some notation. For each T ∈ S^{f},

v_{T} is the vertex of K corresponding to T
F_{T}^{∗} is the dual face of T in K
Y_{T} is the nonsingular part of Σ_{T} × Σ_{T} , and
Q_{T} = Y_{T} /W_{T} .

(3.1) A decomposition of Q. Consider the decomposition of K into dual faces {F_{T}^{∗}}_{T ∈S^{f}} and its pullback to a decomposition {p^{−1}(F_{T}^{∗})} of Q. The purpose of this subsection is to prove the following.

Proposition 3.1.1. Suppose T ∈ S^{f}. Then p^{−1}(F_{T}^{∗}) satisfies the following.

(i) p^{−1}(F_{T}^{∗}) is homotopy equivalent to Q_{T}. Hence, p^{−1}(F_{T}^{∗}) is aspherical with fundamental group A_{T}.

(ii) If T′ ⊆ T, then the map of fundamental groups A_{T′} → A_{T} induced by the inclusion p^{−1}(F_{T′}^{∗}) ⊆ p^{−1}(F_{T}^{∗}) is the natural homomorphism (up to conjugacy).

(iii) The inclusion p^{−1}(F_{T}^{∗}) ↪ Q induces an injection on fundamental groups.

Proof. (i) Each component of the inverse image of F_{T}^{∗} in Σ (under the orbit map Σ → K) is isomorphic to Σ_{T}. Hence,

p^{−1}(F_{T}^{∗}) = ((Σ_{T} × Σ) ∩ Y)/W_{T}

By Lemma 2.2.6, the right hand side is homotopy equivalent to Y_{T} /W_{T} (= Q_{T}). By Corollary 2.2.5, Q_{T} is homotopy equivalent to Y_{T} /W_{T}, where Y_{T} is the complex hyperplane complement corresponding to the orthogonal reflection group W_{T} on ℝ^{T}. By Deligne’s Theorem [D, Théorème 4.4], Y_{T} /W_{T} is aspherical with fundamental group A_{T}, which proves the second sentence of (i).

(ii) It is clear from Deligne’s proof that for T′ ⊂ T, any map Y_{T′}/W_{T′} → Y_{T} /W_{T} induced by a T′-face-preserving map of fundamental chambers induces the standard homomorphism A_{T′} → A_{T}. Since F_{T′}^{∗} ↪ F_{T}^{∗} is T′-face preserving, we have a diagram

p^{−1}(F_{T′}^{∗}) ⟶^{∼} Y_{T′}/W_{T′}
↓  ↓
p^{−1}(F_{T}^{∗}) ⟶^{∼} Y_{T} /W_{T}

in which all arrows are induced by T′-face-preserving maps of fundamental chambers. By Lemma 2.2.2(i) such maps are unique up to homotopy; hence, the diagram commutes up to homotopy. So, (ii) holds.

(iii) Let

Z_{T} = (Σ_{T} × Σ) ∩ Y

= (Σ_{T} × Σ) − ⋃_{r∈W_{T}} P_{r} × P_{r}.

where the union is over all reflections r in W_{T}. In other words, Z_{T} is a component of the inverse image of p^{−1}(F_{T}^{∗}) in Y, (so, Z_{T} /W_{T} = p^{−1}(F_{T}^{∗})). By Lemma 2.2.6, the inclusion

Z_{T} ↪ Σ × Σ − ⋃_{r∈W_{T}} P_{r} × P_{r}

is a W_{T} -equivariant homotopy equivalence. Since the inclusion factors through Y, the induced map Z_{T} /W_{T} ↪ Y/W_{T} gives rise to an injection on fundamental groups. The same holds for Z_{T} /W_{T} ↪ Y/W, since Y/W_{T} → Y/W is the projection map of a covering space. □

(3.2) A thickened version of Q. We would like to say that Proposition 3.1.1 means that p : Q → K is a complex of spaces over K with A(K) as its associated complex of groups. Unfortunately, this is not quite true. The problem is that, although the inverse images of the dual cells are correct, the inverse images of the vertices are not. To remedy this, we define Q̂, a “thickened” version of Q.

Recall that the dual face F_{T}^{∗} is the geometric realization of (S_{≤T}^{f})′ and that F^{∘} _{T}^{∗} =

|(S_{≤T}^{f})′| − |(S_{<T}^{f})′|. Each point x in K belongs to a unique “open” dual face F^{∘} _{T}^{∗}, for some T ∈ S^{f}.

The space Q̂ is the subspace of K × Q defined by

Q̂ = {(x, q) ∈ K × Q|x ∈ F^{∘} _{T}^{∗} ⇒ q ∈ p^{−1}(F_{T}^{∗})}.

There is a projection map p̂: Q̂ → K defined by p̂(x, q) = x. Choose a basepoint q_{0} ∈ Q lying over the central vertex v_{∅} of K. Since v_{∅} belongs to each dual face, the point (x, q_{0}) lies in Q̂ for all x ∈ K. Hence, the map s : K → Q̂ defined by x → (x, q_{0}) is a well-defined section of p̂.

Theorem 3.2.1. p̂: Q̂ → K is an aspherical realization of A(K).

Proof. First we show that p̂ : Q̂ → K is a simple complex of spaces over K. (See §1.4.). Thus, we must define, for each T ∈ S^{f}, a deformation retraction r_{T} : p̂^{−1}(F_{T}^{∗}) → p̂^{−1}(v_{T}) such that r_{T} ∘ s is constant on the 1-skeleton of F_{T}^{∗}. We have

p̂^{−1}(v_{T}) = v_{T} × p^{−1}(F_{T}^{∗})
p̂^{−1}(F_{T}^{∗}) ⊆ F_{T}^{∗} × p^{−1}(F_{T}^{∗}).

The map r_{T}, defined by r_{T} (x, q) = (v_{T} , q) is clearly a deformation retraction (since F_{T}^{∗} is contractible to v_{T}) and on F_{T}^{∗}, r_{T} ∘ s is the constant map x → (v_{T} , q_{0}). Thus, Q̂ is a complex of spaces over K.

It follows from Proposition 3.1.1 that the associated complex of groups is A(K) (since the fundamental group of p̂^{−1}(v_{F}) = p^{−1}(F_{T}^{∗}) is A_{T}) and that it is an aspherical realization of A(K) (since p^{−1}(F_{T}^{∗}) is aspherical). □

Corollary 3.2.2.

(i) π_{1}(Q̂) = A

(ii) Q̂ is homotopy equivalent to BA(K).

Proof. Strictly speaking, to deduce this from Theorem 3.2.1 one should have that the complex of spaces p̂: Q̂ → K is cellular, that is, that the fibers of p̂ are CW - complexes. This will be true only after a slight modification of Q̂. One replaces Q̂ by a slightly smaller homotopy equivalent space which is a subcomplex of a fine subdivision of K × Σ × Σ. The details are left to the reader. □

There is a natural inclusion i : Q → Q̂ defined by i(q) = (p(q), q) and a retraction r : Q̂ → Q given by r(x, q) = q.

Proposition 3.2.3. Q is a deformation retract of Q̂.

Proof. We have i ∘ r(x, q) = (p(q), q) and the points x and p(q) both belong to F_{T}^{∗} if

x ∈ F^{∘} _{T}^{∗}. We construct a homotopy from the identity to i ∘ r by deforming x to p(q) in F_{T}^{∗}. As we shall explain in detail in §4.3, the subcomplex F_{T}^{∗} can be canonically identified with a regular Euclidean cube (of dimension Card (T)) so that if T′ ⊂ T, then F_{T′}^{∗} corresponds to a face of the cube for F_{T}^{∗}. (F_{T}^{∗} is simplicially isomorphic to the cone on the barycentric subdivision of the simplex on T and this complex is isomorphic to a standard subdivision of the cube.) In any case, since F_{T}^{∗} can be canonically identified with a convex subset of Euclidean space, for any x_{1}, x_{2} ∈ F_{T}^{∗} and t ∈ [0, 1], tx_{1} + (1 − t)x_{2} is a well-defined point in F_{T}^{∗}. Hence,

r_{t}(x, q) = (tp(q) + (1 − t)s, q)

is a well-defined homotopy from the identity to i ∘ r. □

Corollary 3.2.4.

(i) π_{1}(Q) = A

(ii) Q is homotopy equivalent to BA(K).

Corollary 3.2.5. (van der Lek [L, Theorem 4.13]). A(K) is developable. (In other words, for each T ∈ S^{f}, the standard homomorphism A_{T} → A is injective).

Proof. This is immediate from Propostion 3.1.1 (iii) and part (i) of the previous Corollary. □

¿From Theorem 1.5.1 (ii) and Corollary 3.2.4 (ii), we get the following.

Corollary 3.2.6. Q is aspherical if and only if Φ is contractible.

By (2.2.5), Y/W is homotopy equivalent to Q (where Y denotes the hyperplane complement). Hence, the above corollary reduces the K(π, 1)-problem to the problem of showing that Φ is contractible.

Remark. The space Q̂ is an artifact of our desire to frame the above results in the context of [H2]; it could have been dispensed with. An alternative argument would be to consider the universal covering space Ỹ of Y (Ỹ is also the universal covering of Q) and the open cover of Ỹ by the path components of the inverse images of open stars of vertices in K. Using the arguments in [D, §3] one can then show that each open set in this cover, as well as each nonempty intersection of such open sets, is contractible and that the nerve of this open cover is Φ. It follows that π_{1}(Q) = A, that Φ×_{A} Ỹ is homotopy equivalent to BA(K) and hence, as before, that Q is aspherical if and only if Φ is contractible.

§4. Metrics on Φ.

Gromov [G] and Moussong [M] defined piecewise Euclidean metrics on the modified Coxeter complex. In this section we discuss the analogous metrics for the modified Deligne complex Φ. We show that, under the hypotheses of Theorem A of the Introduction, Gromov’s metric on Φ satisfies CAT (0) and hence, Φ is contractible. Conjecturally, Moussong’s metric is always CAT (0).

(4.1) CAT (r)-metrics. This terminology was introduced by Gromov in [G]. (“CAT ” stands for “comparison of Alexandrov and Topogonov”.) The idea is to define a notion for singular metric spaces which corresponds to “sectional curvature bounded above by r” for Riemannian manifolds. The theory was developed, in a context applicable to our situation, by Bridson in [B].

Let (X, d) be a metric space. A geodesic segment in X is the image of an isometric embedding γ : [a, b] → X of an interval into X. One says that (X, d) is a geodesic space (or a length space) if any two points in the same path component of X can be connected by a geodesic segment.

As r is positive, zero or negative, let 𝕄_{r}^{n} stand for, respectively, the n-sphere of radius 1/\sqrt{r}, Euclidean n-space, or hyperbolic n-space of curvature r.

A geodesic space (X, d) has a CAT (r)-metric (or is a CAT (r)-space) if for any geodesic triangle in X, the distance from a vertex to a point on the opposite side is less than or equal to the distance between the corresponding points on a comparison triangle in 𝕄_{r}^{2}.

If X is a connected CAT (r)-space with r ≤ 0, then there is a unique geodesic segment connecting any two points in X and such geodesic segments vary continuously with the endpoints. It follows that X is contractible.

An 𝕄_{r}-structure on a cell complex X means an identification of each cell of X with a convex cell in 𝕄_{r}^{n}, for some n, so that common faces are congruent. If r = 0 (resp., r = 1), then we call such a structure piecewise Euclidean (resp.,piecewise spherical). If the number of isometry types of cells in the 𝕄_{r}-structure is finite, the we say it satisfies the finite shape condition (F S).

An 𝕄_{r}-structure defines an intrinsic metric on X, by taking d(x, y) to be the infimum of the lengths of paths from x to y which are piecewise geodesic in the sense that they are geodesic in each cell. If X is locally finite or if it satifies (F S), then Moussong [M] and Bridson [B] have shown that this infimum is actually attained. Hence, (X, d) is a geodesic space. The question of whether d is a CAT (r)-metric is more subtle and is discussed in the next subsection.

(4.2) Link conditions. We assume that all our cell complexes are finite dimensional.

Given an 𝕄_{r}-structure on X it is possible to determine if the resulting intrinisc metric is CAT (r) by looking at links of vertices in X.

For a vertex v of a convex cell σ in 𝕄_{r}^{n}, define Lk(v, σ) as the set of inward pointing unit tangent vectors at v; it is naturally a convex cell in a standard sphere. It follows that

Lk(v, X) = ⋃_{v∈σ} Lk(v, σ)

is a piecewise spherical cell complex.

The following theorem was stated in [G, 4.2.A, p. 120] and proved under the (F S) hypothesis in [B, p. 396].

Theorem 4.2.1. (Gromov, Bridson). Suppose that X is a cell complex with 𝕄_{r}-structure, that r ≤ 0, that X is simply connected and that it satisfies (F S). Then X is a CAT (r)- space if and only if Lk(v, X) is a CAT (1)-space for every vertex v in X.

In general, proving that links are CAT (1)-spaces is no easy matter. However, there is one situation in which the problems can be reduced to a combinatorial one: when the cells of Lk(v, X) are “all right” simplices.

Definition. An n-simplex σ in 𝕊^{n} is all right if each of its edge lengths is \frac{π}{2} . (In other words, σ is isometric to the intersection of 𝕊^{n} with the simplicial cone in ℝ^{n+1} spanned by the standard basis.)

Remark. The link of a vertex in a regular Euclidean cube is an all right simplex. Hence, if X is piecewise Euclidean and the cells are regular cubes (of varying dimensions), then each cell of Lk(v, X) is an all right simplex.

Definition. A simplicial complex L is a flag complex if whenever {v_{0}, ⋯, v_{k}} is a set of distinct vertices which are pairwise joined by edges, then {v_{0}, ⋯, v_{k}} spans a k-simplex in L. (In other words, L is “determined by its 1-skeleton”.)

Proposition 4.2.2. (Gromov [G, p. 122]). A piecewise spherical simplicial complex made up of all right simplices is a CAT (1)-space if and only if it is a flag complex.

Corollary 4.2.3. A simply connected, piecewise Euclidean complex made up of regular Euclidean cubes is a CAT (0)-space if and only if the link of each vertex is a flag complex.

Remark. It is not necessary to assume, in Proposition 4.2.2, that the simplicial complex be finite or even locally finite. In fact, since a geodesic segment in an all right simplicial complex must lie in some finite subcomplex, we see that the general case of Proposition 4.2.2 follows from the special case where the complex is finite. A similar remark applies to Proposition 4.2.4, below.

Proposition 4.2.2 was generalized by G. Moussong [M] as follows.

Definition. A spherical simplex σ in 𝕊^{n} has size ≥ \frac{π}{2} if each of its edge lengths is at least \frac{π}{2} .

Definition. A piecewise spherical simplicial complex L is a metric flag complex if whenever {v_{0}, ⋯, v_{k}} is a set of distinct vertices such that the v_{i} are pairwise joined by edges and such that there exists a k-simplex in 𝕊^{k} with the same edge lengths, then {v_{0}, ⋯, v_{k}} spans a simplex in L. (In other words, L is “metrically determined by is 1-skeleton”.)

Proposition 4.2.4. (Moussong [M]). A piecewise spherical simplicial complex all of whose simplices have size ≥ \frac{π}{2} is a CAT (1)-space if and only if it is a metric flag complex.

(4.3) Cubical metrics on K, Σ and Φ. As in §1, let (W, S) be a Coxeter system and K the geometric realization of the derived complex of S^{f}. For each T ∈ S^{f}, the dual face,

F_{T}^{∗} = |(S_{≤T}^{f})′|

is the cone on the barycentric subdivision of a simplex (with cone point v_{∅}). This is isomorphic to a standard subdivision of a cube of dimension Card (T). The vertices of the cube are the vertices of K which lie in F_{T}^{∗}, i.e., those of the form v_{T′}, T′ ⊆ T. The cubical faces of F_{T}^{∗} are the subsets of the form F_{T_{1}} ∩ F_{T_{2}}^{∗} where T_{1} ⊆ T_{2} ⊆ T.

F_{{s,t,r}}^{∗} viewed as a cube

Hence, the cubical faces of K which contain v_{T} are those of the form F_{T_{1}} ∩ F_{T_{2}}^{∗}, where T_{1} ⊆ T ⊆ T_{2} and T_{2} ∈ S^{f}.

Identify each such cube (and its translates) with a regular Euclidean cube of edge length 1. This defines a piecewise Euclidean, cubical structure on K, as well as on the modified Coxeter complex Σ and on the modified Deligne complex Φ. We wish to determine when the resulting intrinsic metrics on K, Σ and Φ are CAT (0). By Corollary 4.2.3 this reduces to determining when links of vertices are flag complexes.

Let Δ_{T} be the simplex on T and let Σ̂_{T} and Φ̂_{T} denote, respectively, the (unmodified) Coxeter complex and (unmodified) Deligne complex, i.e.,

Σ̂_{T} = (W_{T} × Δ_{T})/ ∼
Φ̂_{T} = (A_{T} × Δ_{T})/ ∼

Lemma 4.3.1. Let T ∈ S^{f} and v_{T} the corresponding vertex of K. Then, with respect to the cubical structures discussed above, we have the following decomposition of links as joins:

(i) Lk(v_{T} , K) = Δ_{T} ∗ Lk(v_{T} , F_{T})

(ii) Lk(v_{T} , Σ) = Σ̂_{T} ∗ Lk(v_{T} , F_{T})

(iii) Lk(v_{T} , Φ) = Φ̂_{T} ∗ Lk(v_{T} , F_{T}).

Proof. As X = K, Σ or Φ, let G_{T} stand for {1}, W_{T} or A_{T} and let X_{T} stand for F_{T}^{∗}, Σ_{T} or Φ_{T}, respectively. The cubical faces of X containing v_{T} are the G_{T} -translates of the cubes F_{T_{1}} ∩ F_{T_{2}}^{∗} where T_{1} ⊆ T ⊆ T_{2}. Such a cube can be decomposed as a Cartesian product:

F_{T_{1}} ∩ F_{T_{2}}^{∗} = (F_{T_{1}} ∩ F_{T}^{∗}) × (F_{T} ∩ F_{T_{2}}^{∗})
⊆ F_{T}^{∗} × F_{T} .

Thus, the cubes containing v_{T} lie in (G_{T} F_{T}^{∗}) × F_{T} (since G_{T} fixes F_{T}). Since G_{T} F_{T}^{∗} is naturally isometric to X_{T} it follows that

Lk(v_{T} , X) = Lk(v_{T} , X_{T}) ∗ Lk(v_{T} , F_{T}).

Since Lk(v_{T} , X_{T}) can be identified with Δ_{T} , Σ̂_{T} or Φ̂_{T} as X_{T} = F_{T}^{∗}, Σ_{T} orΦ_{T}, the result follows. □

The join A ∗ B of two simplicial complexes is a flag complex if and only if both A and B are flag complexes. Thus we need to analyze the terms on the right hand sides of the formulas in Lemma 4.3.1 separately. The simplex Δ_{T} is obviously a flag complex. So is Σ̂_{T}. (This follows easily from its description as a poset of cells arising from a finite reflection group, cf., [Br, p. 29].) The following lemma is the main technical result needed to prove that Φ is contractible. Its proof occupies §4.5.

Lemma 4.3.2. Φ̂_{T} is a flag complex for all T ∈ S^{f}.

On the other hand, we shall see that Lk(v_{T} , F_{T}) is not always a flag complex.

We recall that S_{>∅}^{f} is an abstract simplicial complex: its vertex set is S and a subset T of S is a simplex if and only if T ∈ S_{>∅}^{f}.

Lemma 4.3.3.

(i) Lk(v_{∅}, F_{∅}) = S_{>∅}^{f}

(ii) Lk(v_{T} , F_{T}) = Lk(T, S_{>∅}^{f}), for T ∈ S_{>∅}^{f}.

Proof. The cubes of F_{T} which properly contain v_{T} are intersections of the form F_{T} ∩ F_{T′}, where T < T′, T′ ∈ S_{>∅}^{F}, and such a cube contributes a simplex of dimension Card (T′ − T) − 1 to Lk(v_{T} , F_{T}). The lemma follows. □

Consider the following condition on a Coxeter system (W, S).

(F C):If T ⊆ S and every pair of elements in T generates a finite subgroup of W, then T generates a finite subgroup of W.

Lemma 4.3.4. The following statements are equivalent.

(1) Lk(v_{T} , F_{T}) is a flag complex for all T ∈ S^{f}.

(2) S_{>∅}^{f} is a flag complex

(3) (W, S) satisfies (F C).

Proof. If a simplicial complex is a flag complex, then so is the link of each of its simplices. Hence, using Lemma 4.3.3, we see that (1) and (2) are equivalent. The equivalence of (2) and (3) is immediate. □

Theorem 4.3.5. Let X stand for K, Σ or Φ with the piecewise Euclidean, cubical structure described above and let d_{□} be the resulting intrinsic metric on X. Then (X, d_{□}) is a CAT (0)-space if and only if (W, S) satisfies (F C).

Proof. The spaces K, Σ and Φ are simply connected (by Theorem 1.5.1). So, by Corollary 4.2.3, (X, d_{□}) is a CAT (0)-space if and only if the link of each vertex in X is a flag complex. Hence, the theorem follows from Lemmas 4.3.1, 4.3.2 (to be proved in the next subsection) and 4.3.4. □

Since a CAT (0)-space is contractible, we have, as a corollary, the following theorem which was the main goal of this section.

Theorem 4.3.6. If (W, S) satisfies (F C), then Φ is contractible.

This theorem, together with Corollary 2.2.5 and Corollary 3.2.6, proves Theorem A of the Introduction.

Remark. Suppose that (F C) fails for (W, S). Let T be a minimal subset of S such that (F C) fails for (W_{T} , T). Then W_{T} is infinite, Card (T) ≥ 3, and every proper subset of T generates a finite Coxeter group. Given the list of diagrams of finite Coxeter groups (e.g. [Bo, Ch VI, §4,.1, Théoremè 1]), it is a simple matter to write down the diagrams of all such W_{T} and this was done by Lanner in [La]. His well-known lists are repeated below in the following two tables. It turns out that any such W_{T} can be realized as a group generated by isometric reflections on Euclidean space or hyperbolic space of dimension Card (T) − 1.

Table 1 Euclidean Coxeter Groups with fundamental chamber an n-simplex , n ≥ 2

Table 2 Hyperbolic Coxeter Groups with fundamental chamber an n-simplex

The previous remark proves the following.

Lemma 4.3.7. (W, S) satisfies (F C) if and only if for all T ⊆ S, the diagram of (W_{T} , T) does not appear in Table 1 or 2, above.

Remark. Theorem 4.3.6 also can be proved by a simple inductive argument, without showing that it admits a CAT (0) metric. Indeed, if W is finite, then Φ is a cone and therefore, contractible. If W is infinite and satisfies (F C), then there exist elements t_{1}, t_{2} in S so that t_{1}t_{2} is of infinite order in W. Let T_{1} = S − {t_{1}}, T_{2} = S − {t_{2}} and T_{0} = S − {t_{1}, t_{2}}. Then K = K_{T_{1}} ∪ K_{T_{2}} and K_{T_{1}} ∩ K_{T_{2}} = K_{T_{0}}, where K_{T_{i}} denotes |(S_{≤T_{i}}^{f})′|, i = 0, 1, 2. Each component of the inverse image of K_{T_{i}} in Φ is isomorphic to Φ_{T_{i}}. Since A is the amalgamated free product of A_{T_{1}} and A_{T_{2}} along A_{T_{0}}, the intersection of a component of type Φ_{T_{1}} and one of type Φ_{T_{2}} is either empty or isomorphic to Φ_{T_{0}}, and the nerve of the covering of Φ by these components is the tree associated to the amalgamated product. By induction, Φ_{T_{i}}, i = 0, 1, 2, is contractible and the theorem follows.

(4.4) The Moussong metric. In this subsection we describe a different piecewise Euclidean structure on X = K, Σ or Φ. Combinatorially, the cell structure on X will be the same as before: the cells are the dual faces and they are combinatorial cubes. However, each such combinatorial cube will be identified with a different convex cell in Euclidean space (called a “Coxeter block”, below). The resulting intrinsic metric on X will be denoted by d_{M}.

By Lemma 4.3.1, there is a simplicial isomorphism

Lk(v_{T} , X) ≅ X̂_{T} ∗ Lk(v_{T} , F_{T}),

where X̂_{T} stands for Δ_{T} , Σ̂_{T} of Φ̂_{T}. In fact, this isomorphism will be an isometry, if ∗ is interpreted as denoting the “orthogonal join” of two piecewise spherical complexes (as defined, for example, in the Appendix of [CD1]). Moreover, the metric d_{M} is designed so that the Coxeter complex Σ̂_{T}, with the induced piecewise spherical structure, will be isometric to the round sphere 𝕊^{Card (T)−1}.

Suppose that (W, S) is a Coxeter system, that S = {s_{1}, ⋯, s_{n}}, and that (m_{ij}) is the associated Coxeter matrix. The associated cosine matrix (c_{ij}) is defined by

c_{ij} = − cos(π/m_{ij})

(where π/∞ is interpreted to be 0).

Suppose (c_{ij}) is positive definite. Choose a basis {u_{i}} for ℝ^{n}, n = Card (S), so that u_{i} · u_{j} = c_{ij}. We can then realize W as a linear reflection group on ℝ^{n} by letting s_{i} act as the orthogonal reflection across the hyperplane (u_{i})^{⊥}. A fundamental chamber is the simplicial cone

C_{S} = {x ∈ ℝ^{n}|x · u_{i} ≤ 0, ∀i}.

It follows that W is a discrete subgroup of O(n) and hence, finite. Conversely, if W is finite, then by representing it as a linear reflection group and choosing an invariant inner product we see that its cosine matrix must be positive definite. Thus, W is finite if and only if (c_{ij}) is positive definite (see [Bo, Ch. V, §4.8, Théoremè 2]). The Coxeter complex is then naturally identified with 𝕊^{n−1}, triangulated by the W -translates of the spherical simplex

σ_{S} = C_{S} ∩ 𝕊^{n−1}.

Let {v_{i}} denote the dual basis to {u_{i}} and let C_{S}^{∗} be the dual simplicial cone,

C_{S}^{∗} = {x ∈ ℝ^{n}|x · v_{i} ≥ 0, ∀i}.

Equivalently, C_{S}^{∗} is the cone of all nonnegative linear combinations of the u_{i}. Let σ_{S}^{∗} be the dual spherical simplex,

σ_{S}^{∗} = C_{S}^{∗} ∩ 𝕊^{n−1}.

The vertex set of σ_{S}^{∗} is {u_{i}} and the length of the edge from u_{i} to u_{j} is cos^{−1}(c_{ij}) = π − π/m_{ij}. Hence, σ_{S}^{∗} has size ≥ π/2 in the sense of §4.2.

Let x_{0} ∈ C_{S} be the point defined by x_{0} · u_{i} = −1, i = 1, ⋯ n. The Coxeter block for (W, S) is the Euclidean polytope,

B_{S} = C_{S} ∩ (x_{0} + C_{S}^{∗})

Its faces are the subsets B_{T_{1},T_{2}}, for T_{1} ⊆ T_{2} ⊆ S, defined by,

B_{T_{1},T_{2}} = {x ∈ B_{S}|x · u_{i} = 0 for s_{i} ∈ T_{1}and (x − x_{0}) · v_{j} = 0 for s_{j} ∈ T_{2}}

which is a convex polytope of dimension Card (T_{2} −T_{1}). It follows that B_{S} is combinatorially equivalent to an n-cube. For T ⊆ S, the Coxeter block B_{T} can be naturally identified with the face B_{∅,T} of B_{S}.

For T ⊆ S, put x_{T} = B_{T,T} so that {x_{T} } is the vertex set of B_{S}. The faces B_{T,S} and B_{∅,T} are contained in orthogonal faces of C_{S} and x_{0} + C_{S}^{∗}, respectively. Hence,

Lk(x_{T} , B_{S}) = Lk(x_{T} , B_{∅,T}) ∗ Lk(x_{T} , B_{T,S})
= σ_{T} ∗ σ_{S−T}^{∗}

(where ∗ denotes the orthogonal join). In particular, Lk(x_{S}, B_{S}) = σ_{S} and Lk(x_{∅}, B_{S}) = σ_{S}^{∗}.

Now we return to the situation in which (W, S) can be infinite. As in §4.3, we view the fundamental chamber K as a union of combinatorial cubes F_{T}^{∗}, T ∈ S^{f}. Following [M] (or [CD2, Section 6]), we define a piecewise Euclidean structure on K by identifying F_{T}^{∗} with B_{T} in such a fashion that for T_{1} ≤ T_{2} ≤ T, the face F_{T_{1}} ∩ F_{T_{2}}^{∗} of F_{T}^{∗} is identified with B_{T_{1},T_{2}}. In particular, the vertex v_{T} (= F_{T} ∩ F_{T}^{∗}) is identified with x_{T}. Similarly, we get piecewise Euclidean structures on Σ and Φ.

Next we consider the induced piecewise spherical structure on links. Recall that, as an abstract simplicial complex, Lk(v_{∅}, F_{∅}) = S_{>∅}^{f}. Let

K_{0} = |S_{>∅}^{f}|

with the piecewise spherical structure obtained by identifying the simplex corresponding to T with the spherical simplex σ_{T}^{∗}. As was previously noted, each such simplex has size ≥ π/2. Also, T ∈ S_{>∅}^{f} ⇔ W_{T} is finite ⇔ the cosine matrix for W_{T} is positive definite ⇔ {π − π/m_{ij}}_{s_{i},s_{j}∈T} is the set of edge lengths of a spherical simplex. Hence, K_{0} is a metric flag complex. Since the property of being a metric flag complex is inherited by links (cf., [M] or [CD 2, Lemma 2.41]) we get the following lemma as a corollary to Proposition 4.2.4.

Lemma 4.4.1. (Moussong [M]). K_{0} and the links of all simplices in K_{0} are CAT (1)- spaces.

Definition. Suppose W_{T} is finite. The natural piecewise spherical structures on the Coxeter complex Σ̂_{T} and the Deligne complex Φ̂_{T} are defined by identifying the fundamental simplex Δ_{T} with the spherical simplex σ_{T} (= C_{T} ∩ 𝕊^{Card (T)−1}).

Arguing as in Lemmas 4.3.1 and 4.3.3, we get the following.

Lemma 4.4.2. Let X = K, Σ or Φ with the piecewise Euclidean structure of Moussong discussed above. For T ∈ S^{f}, let X̂_{T} = σ_{T} , Σ̂_{T} or Φ̂_{T} (as X = K, Σ or Φ) with its natural piecewise spherical structure. Then the link of any vertex of X can be decomposed as an orthogonal join as follows:

Lk(v_{T} , X) = X̂_{T} ∗ Lk(v_{T} , F_{T}).

Moreover, metrically, Lk(v_{∅}, F_{∅}) = K_{0} and, for T ≠ ∅, Lk(v_{T} , F_{T}) = Lk(σ_{T}^{∗} , K_{0}).

The orthogonal join of two piecewise spherical polyhedra is CAT (1) if and only if each of them is CAT (1) (by Lemmas A2 and A6 of the Appendix of [CD 1]). Hence, it only remains to decide when X̂_{T} is a CAT (1)-space. Since Σ̂_{T} is isometric to the round sphere, it is CAT (1). This gives the following result of [M].

Theorem 4.4.3. (Moussong). (Σ, d_{M}) is is CAT (0).

Conjecture 4.4.4. (Φ, d_{M}) is CAT (0).

As we have seen, this conjecture is equivalent to Conjecture 3 of the Introduction which we restate here.

Conjecture 3. For any finite Coxeter system (W_{T} , T), the Deligne complex Φ̂_{T} with its natural piecewise spherical structure is CAT (1).

Conjecture 3 is true when W_{T} is right-angled (i.e., when W_{T} = (ℤ/2)^{T}), for in this case the natural piecewise spherical structure on Φ̂_{T} coincides with the all right structure of §4.3. There is one other case when we know the conjecture is true.

Proposition 4.4.5. Conjecture 3 is true when Card (T) = 2 (so that W_{T} is the dihedral group D_{2m} of order 2m).

Proof. In this case Φ̂_{T} is a 1-dimensional complex with all edges of length π/m. Hence, it is CAT (1) if and only if it has no circuits with less than 2m edges. This condition holds by [AS, Lemma 6, p. 210]. □

Thus, Conjecture 4.4.3 holds for W such that Card (T) ≤ 2 for all T ∈ S^{f} (for example, if W is of “large type” as defined in [AS]). Theorem B of the Introduction is an immediate consequence.

(4.5) The Proof of Lemma 4.3.2. Throughout this subsection, A is an Artin group of finite type. Our goal is to prove that the Deligne complex Φ̂ associated to A is a flag complex.

Before proving this we need a number of preliminary lemmas about Artin groups of finite type. These are based on work in [C1] and [C2], which in turn is based on ideas of Deligne [D] and Garside [Ga].

Let F^{+}(S) be the free monoid on S and let A^{+} be the quotient of F^{+}(S) by the equivalence relation generated by: prod(s_{i}, sj; m_{ij}) = prod(s_{j}, s_{i}, m_{ij}), for s_{i}, s_{j} ∈ S. In other words, if ⟨S|R⟩ is the standard presentation for A, then A^{+} is the monoid with monoid presentation ⟨S|R⟩. For example, when A is the braid group, A^{+} is the monoid of positive braids.

The natural map A^{+} → A is an injection; hence, we can identify A^{+} with a subset of A. Moreover, for any a ∈ A^{+}, we can write a = bc^{−1} for some b, c ∈ A^{+}. If T ⊆ S, then the natural map A_{T}^{+} → A^{+} is also an injection (since A_{T} → A is) so we can also view A_{T}^{+} as a subset of A.

Lemma 4.5.1.

(i) Suppose that a, b, c are in A^{+}, that a = bc, and that a ∈ A_{T}^{+}, then b and c both lie in A_{T}^{+}.

(ii) A_{T}^{+} = A_{T} ∩ A^{+}

Proof. (i) Choose representative words â ∈ F^{+}(T) and b̂, ĉ ∈ F^{+}(S) for a, b and c. If a = bc in A^{+} , then â is related to b̂ĉ via the equivalence relation generated by the

prod(s_{i}, s_{j}; m_{ij}) = prod(s_{j}, s_{i}; m_{ij}).

Under this relation, no generator can be completely eliminated. Hence, if b̂ or ĉ involves some s_{i}, s_{i} ∉ T, then so does â.

(ii) Clearly, A_{T}^{+} ⊂ A_{T} ∩ A^{+}. Conversely, let a ∈ A_{T} ∩ A^{+}. Write a = bc^{−1} for b, c ∈ A_{T}^{+}. Then ac = b in A^{+}, so by part (i), a lies in A_{T}^{+}. □

There are two partial orderings on A^{+}, denoted <_{ℓ} and <_{r}, and defined by:

a <_{ℓ} b, if ac = b for some c ∈ A^{+}, and
a <_{r} b, if ca = b for some c ∈ A^{+}.

Recall the section m : W → A defined in §1.1. Let w_{ℓ} denote the (unique) element of longest length in W and let Δ = m(w_{ℓ}). Then Δ^{2} is central in A and for any a ∈ A, Δ^{k}a and aΔ^{k} lie in A^{+} for k sufficiently large.

Let M = m(W). By definition of m, M ⊂ A^{+}. In fact, we can describe M in terms of the partial orderings on A^{+} as follows,

M = {a ∈ A^{+}|a <_{ℓ} Δ}
= {a ∈ A^{+}|a <_{r} Δ}.

Since S ⊂ M, M generates A^{+} as a monoid. If M^{k} denotes the set of elements in A^{+} which can be written as a product of at most k elements of M, then it follows from [C2, Remark 2.4 and Lemma 3.4] that

M^{k} = {a ∈ A^{+}|a <_{ℓ} Δ^{k}}
= {a ∈ A^{+}|a <_{r} Δ^{k}}.

Lemma 4.5.2. Let a, b ∈ A^{+} and let ∗ = ℓ or r.

(i) {c ∈ A^{+}|c <_{∗} a and c <_{∗} b} has a unique maximal element denoted glb_{∗}(a, b).

(ii) {c ∈ A^{+}|a <_{∗} c and b <_{∗} c} has a unique minimal element denoted lub_{∗}(a, b).

(iii) If T ⊆ S, then {c ∈ A^{+}|c <_{∗} a} ∩ A_{T}^{+} has a unique maximal element, denoted glb_{∗}(a, A_{T}^{+}).

Proof. By symmetry, it suffices to prove each part of the lemma for either ∗ = ℓ or ∗ = r. (i) For ∗ = r, this is precisely Corollary 2.7 of [C2]. (ii) Choose k large enough so that a, b ∈ M^{k}, hence a <_{ℓ} Δ^{k} and b <_{ℓ} Δ^{k}. Let

a_{k} = a^{−1}Δ^{k} and b_{k} = b^{−1}Δ^{k}. Then a_{k} ∈ A^{+} and a_{k} <_{r} Δ^{k} and similarly for b_{k}. Consider the posets

L_{k} = {c ∈ A^{+}|a <_{ℓ} c, b <_{ℓ} c, c <_{ℓ} Δ^{k}}
R_{k} = {c ∈ A^{+}|c <_{r} a_{k}, c <_{r} b_{k}},

with L_{k} partially ordered by <_{ℓ} and R_{k} partially ordered by <_{r}. The map c → c_{k} = c^{−1}Δ^{k} is an order-reversing bijection from L_{k} to R_{k}. Thus, if d_{k} is the unique maximal element of R_{k} (which exists by part (i)), then d̂_{k} = Δ^{k}d_{k}^{−1} is the unique minimal element of L_{k}. Finally, note that since a_{k+1} = a_{k}Δ and b_{k+1} = b_{k}Δ, we must have d̂_{k+1} = Δ^{k+1}d_{k+1}^{−1} = Δ^{k}d_{k}^{−1} = d̂_{k}. In other words, the sets L_{k} ⊆ L_{k+1} ⊆ L_{k+2} ⊆ ⋯ all have the same unique minimal element, d̂_{k}. Thus, d̂_{k} is also the unique minimal element for

⋃_{j≥k} L_{j} = {c ∈ A^{+}|a <_{ℓ} c, b <_{ℓ} c},

i.e., d̂_{k} = lub_{ℓ}(a, b).

(iii) Let M_{T} = m(W_{T}) = M ∩ A_{T}^{+} and let Δ_{T} be the image of the longest element of W_{T}. It follows from Lemma 4.5.1 (i) that M^{k} ∩A_{T}^{+} = (M_{T})^{k}. Choosing k so that a <_{∗} Δ^{k}, we therefore have that

{c ∈ A^{+}|c <_{∗} a} ∩ A_{T}^{+} ={c ∈ A^{+}|c <_{∗} a} ∩ M^{k} ∩ A_{T}^{+}
={c ∈ A^{+}|c <_{∗} a} ∩ M_{T}^{k}
={c ∈ A^{+}|c <_{∗} a, c < Δ_{T}^{k} }

which has a unique maximal element by part (i). □

Lemma 4.5.3. Let a ∈ A and T ⊆ S. If aA_{T} ∩A^{+} ≠ ∅, then it contains a unique minimal element a_{0} (with respect to <_{ℓ}) and aA_{T} ∩ A^{+} = a_{0}A_{T}^{+}.

Proof. Since a <_{ℓ} b and a ≠ b implies that the word length of a (with respect to S) is strictly less than the word length of b, there are no infinite descending chains in A^{+}. Hence, if aA_{T} ∩ A^{+} is nonempty, it must have minimal elements.

Suppose a_{1} and a_{2} are both minimal in aA_{T} ∩ A^{+}. Then a_{1}^{−1}a_{2} ∈ A_{T}, so we can write a_{1}^{−1}a_{2} = pq^{−1} for p, q ∈ A_{T}^{+}. Thus,

a_{1}p = a_{2}q, p, q ∈ A_{T}^{+}.

Let c_{0} = lub_{r}(p, q) in A_{T}^{+}. Then a_{1}p = a_{2}q = a_{0}c_{0} for some a_{0} ∈ A^{+}. It follows that a_{0}A_{T} = a_{1}A_{T} = a_{2}A_{T} = aA_{T}, i.e., a_{0} ∈ aA_{T} ∩A^{+}. On the other hand, p <_{r} c_{0}, so c_{0} = dp for some d ∈ A^{+}. Thus, a_{0}dp = a_{0}c_{0} = a_{1}p, so a_{0}d = a_{1}. In other words, a_{0} <_{ℓ} a_{1}. The same argument applied to q <_{r} c_{0} shows that a_{0} <_{ℓ} a_{2}. But this contradicts the minimality of a_{1} and a_{2} unless a_{0} = a_{1} = a_{2}. Finally note that if b is any element of aA_{T} ∩ A^{+}, then a_{0} <_{ℓ} b, so by Lemma 4.5.1 (ii), a_{0}^{−1}b ∈ A_{T} ∩ A^{+} = A_{T}^{+}, i.e., aA_{T} ∩ A^{+} = a_{0}A_{T}^{+}. □

Recall that a “special subgroup” of A is a subgroup of the form A_{T} for some T ⊆ S. It follows from [D, Theorem 4.14] that the intersection of two special subgroups is a special subgroup.

Lemma 4.5.4. Suppose, A_{1}, A_{2} and A_{12} = A_{1} ∩ A_{2} are special subgroups of A and that g ∈ A is such that gA_{12} ∩ A^{+} ≠ ∅. If a_{1} and a_{2} are the minimum elements of gA_{1} ∩ A^{+} and gA_{2} ∩ A^{+}, respectively (minimal with respect to <_{ℓ}), then the minimum element of gA_{12} ∩ A^{+} is a = lub_{ℓ}(a_{1}, a_{2}).

Proof. By Lemma 4.5.3, gA_{12} ∩ A^{+} = gA_{1} ∩ gA_{2} ∩ A^{+} = a_{1}A_{1}^{+} ∩ a_{2}A_{2}^{+}. Let b be the minimum element of gA_{12} ∩ A^{+}. Then b = a_{1}c_{1} = a_{2}c_{2} for some c_{1} ∈ A_{1}^{+}, c_{2} ∈ A_{2}^{+}. Thus, b is an upper bound for a_{1} and a_{2}, so a <_{ℓ} b. Write b = ac. Then c <_{r} c_{1} and c <_{r} c_{2}, so by Lemma 4.5.1 (i), c ∈ A_{1}^{+} ∩ A_{2}^{+} = A_{12}^{+}. It follows that a ∈ gA_{12} ∩ A^{+}. This contradicts the minimality of b unless a = b. □

Proof of Lemma 4.3.2. (that Φ̂ is a flag complex): The poset of simplices of Φ̂ is isomorphic to the poset of special cosets aA_{T}, for T ⫋ S, under reverse inclusion. In particular, a k -simplex σ of Φ̂ corresponds to a coset aA_{T} with Card (S −T) = k +1 and Lk(σ, Φ̂) = Φ̂_{T}.

Suppose Φ̂ is not a flag complex. Then there exists a set of vertices {v_{1}, ⋯, v_{k}}, k ≥ 3, of Φ̂ which are pairwise joined by edges, but do not span a simplex of Φ̂. Choosing a minimal such set of vertices, we may assume that every proper subset of {v_{1}, ⋯, v_{k}} spans a simplex. In this case, we say that {v_{1}, ⋯, v_{k}} is an “empty simplex” in Φ̂. It suffices to consider the case k = 3; for if k > 3, and σ is the span of {v_{4}, ⋯, v_{k}}, then {v_{1}, v_{2}, v_{3}} is an “empty triangle” in Lk(σ, Φ̂) = Φ̂_{T}.

A triangle in Φ̂ corresponds to a diagram of special cosets

a_{1}A_{1}  ⊃  a_{12}A_{12}  ⊂  a_{2}A_{2}
a_{13}A_{13}  a_{23}A_{23}
a_{3}A_{3}

where a_{ij}A_{ij} = a_{i}A_{i} ∩ a_{j}A_{j}. To show that this triangle is not “empty” we must show that a_{1}A_{1} ∩ a_{2}A_{2} ∩ a_{3}A_{3} ≠ ∅. After translating by a sufficiently high power of Δ, we may assume that all cosets in the above triangle intersect A^{+}. We can then choose each coset representative a_{i} to be the minimum element of a_{i}A_{i} ∩ A^{+} (by Lemma 4.5.3), and choose a_{ij} to be lub_{ℓ}(a_{i}, a_{j}) which is the minimum element of a_{ij}A_{ij} ∩ A^{+} (by Lemma 4.5.4). Then a_{12} ∈ a_{1}A_{1}^{+} so a_{12} = a_{1}p for some p ∈ A_{1}^{+}. Similarly, a_{13} = a_{1}q for some q ∈ A_{1}^{+}. Set

a = lub_{ℓ}(a_{1}, a_{2}, a_{3})
= lub_{ℓ}(a_{12}, a_{13})
= a_{1}lub_{ℓ}(p, q).

Then a ∈ a_{1}A_{1}^{+}. By symmetry, we must also have a ∈ a_{2}A_{2}^{+} and a ∈ a_{3}A_{3}^{+}. Thus, a_{1}A_{1}^{+} ∩ a_{2}A_{2}^{+} ∩ a_{3}A_{3}^{+} contains the element a and hence, cannot be empty. □

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Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210 charney@math.ohio-state.edu mdavis@math.ohio-state.edu