The Geometry and Topology of Coxeter Groups

Reflection Groups The theory of abstract reflection groups is due to Tits [281]. What is the appropriate notion of an “abstract reflection group”? At first approximation, one might consider pairs (W, S), where W is a group and S is any set of involutions which generates W. This is obviously too broad a notion. Nevertheless, it is a step in the right direction. In Chapter 3, we shall call such a pair a “pre-Coxeter system.” There are essentially two completely different definitions for a pre-Coxeter system to be an abstract reflection group. The first focuses on the crucial feature that the fixed point set of a reflection should separate the ambient space. One version is that the fixed point set of each element of S separates the Cayley graph of (W, S) (defined in Section 2.1). In 3.2 we call (W, S) a reflection system if it satisfies this condition. Essentially, this is equivalent to any one of several well-known combinatorial conditions, e.g., the Deletion Condition or the Exchange Condition. The second definition is that (W, S) has a presentation of a certain form. Following Tits [281], a pre-Coxeter system with such a presentation is a “Coxeter system” and W a “Coxeter group.” Remarkably, these two definitions are equivalent. This was basically proved in [281]. Another proof can be extracted from the first part of Bourbaki [29]. It is also proved as the main result (Theorem 3.3.4) of Chapter 3. The equivalence of these two definitions is the principal mechanism driving the combinatorial theory of Coxeter groups. The details of the second definition go as follows. For each pair (s, t) ∈ S × S, let mst denote the order of st. The matrix (mst) is the Coxeter matrix of (W, S); it is a symmetric S × S matrix with entries in N ∪ {∞}, 1’s on the diagonal, and each off-diagonal entry > 1. Let R := {(st)st}(s,t)∈S×S. August 2, 2007 Time: 12:25pm chapter1.tex INTRODUCTION AND PREVIEW 3 (W, S) is a Coxeter system if 〈S|R〉 is a presentation for W. It turns out that, given any S × S matrix (mst) as above, the group W defined by the presentation 〈S|R〉 gives a Coxeter system (W, S). (This is Corollary 6.12.6 of Chapter 6.) Geometrization of Abstract Reflection Groups Can every Coxeter system (W, S) be realized as a group of automorphisms of an appropriate geometric object? One answer was provided by Tits [281]: for any (W, S), there is a faithful linear representation W ↪→ GL(N,R), with N = Card(S), so that • Each element of S is represented by a linear reflection across a codimension-one face of a simplicial cone C. (N.B. A “linear reflection” means a linear involution with fixed subspace of codimension one; however, no inner product is assumed and the involution is not required to be orthogonal.) • If w ∈ W and w = 1, then w(int(C)) ∩ int(C) = ∅ (here int(C) denotes the interior of C). • WC, the union of W-translates of C, is a convex cone. • W acts properly on the interior I of WC. • Let Cf := I ∩ C. Then Cf is the union of all (open) faces of C which have finite stabilizers (including the face int(C)). Moreover, Cf is a strict fundamental domain for W on I. Proofs of the above facts can be found in Appendix D. Tits’ result was extended by Vinberg [290], who showed that for many Coxeter systems there are representations of W on RN , with N < Card(S) and C a polyhedral cone which is not simplicial. However, the poset of faces with finite stabilizers is exactly the same in both cases: it is the opposite poset to the poset of subsets of S which generate finite subgroups of W. (These are the “spherical subsets” of Definition 7.1.1 in Chapter 7.) The existence of Tits’ geometric representation has several important consequences. Here are two: • Any Coxeter group W is virtually torsion-free. • I (the interior of the Tits cone) is a model for EW, the “universal space for proper W-actions” (defined in 2.3). Tits gave a second geometrization of (W, S): its “Coxeter complex” . This is a certain simplicial complex with W-action. There is a simplex σ ⊂ with dim σ = Card(S) − 1 such that (a) σ is a strict fundamental domain and (b) the elements of S act as “reflections” across the codimension-one faces August 2, 2007 Time: 12:25pm chapter1.tex