# The Geometry and Topology of Coxeter Groups. (LMS-32)

Michael W. Davis
Pages: 600
https://www.jstor.org/stable/j.ctt1r2fnf

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Preface
(pp. xiii-xvi)
Mike Davis
4. Chapter One INTRODUCTION AND PREVIEW
(pp. 1-14)

Finite groups generated by orthogonal linear reflections on$\mathbb{R}^{n}$play a decisive role in

the classification of Lie groups and Lie algebras;

the theory of algebraic groups, as well as, the theories of spherical buildings and finite groups of Lie type;

the classification of regular polytopes (see [69, 74, 201] or Appendix B).

Finite reflection groups also play important roles in many other areas of mathematics, e.g., in the theory of quadratic forms and in singularity theory. We note that a finite reflection group acts isometrically on the unit sphere$\mathbb{S}^{n-1}$of$\mathbb{R}^{n}$.

There is a similar theory of...

5. Chapter Two SOME BASIC NOTIONS IN GEOMETRIC GROUP THEORY
(pp. 15-25)

In geometric group theory we study various topological spaces and metric spaces on which a groupGacts. The first of these is the group itself with the discrete topology. The next space of interest is the “Cayley graph.” It is a certain one dimensional cell complex with aG-action. Its definition depends on a choice of a set of generatorsSforG. Cayley graphs forGcan be characterized asG-actions on connected graphs which are simply transitively on the vertex set (Theorem 2.1.1). Similarly, one can define a “Cayley 2-complex” forGto be any simply connected,...

6. Chapter Three COXETER GROUPS
(pp. 26-43)

Given a groupWand a setSof involutory generators, when does (W, S) deserve to be called an “abstract reflection group” (or a “Coxeter system”)? In this chapter we give the two answers alluded to in 1.1. The first is that for eachs ∈ Sits fixed set separates Cay(W, S) (see 3.2). The second is thatWhas a presentation of a certain form (see 3.3). The main result, Theorem 3.3.4, asserts that these answers are equivalent. Along the way we find three combinatorial conditions (D), (E), and (F) on (W, S), each of which is...

7. Chapter Four MORE COMBINATORIAL THEORY OF COXETER GROUPS
(pp. 44-62)

Except in Section 4.8, (W, S) is a Coxeter system. Aspecialsubgroup ofWis one that is generated by a subset ofS. In 4.1 we show, among other things, that special subgroups are Coxeter groups (Theorem 4.1.6). In 4.3 we show there is a unique element of minimum length in each coset of a special subgroup. This allows us to discuss, in 4.5, certain “convex subsets” ofWsuch as “halfspaces” and “sectors.” In 4.6 we prove that each finite Coxeter group has a unique element of longest length. We use this in proving an important result,...

8. Chapter Five THE BASIC CONSTRUCTION
(pp. 63-71)

Given a Coxeter system (W, S), a spaceXand a family of subspaces$(X_{s})_{s\in S}$, there is a classical construction of a space$\mathcal{U}(W, X)$withW-action.$\mathcal{U}(W, X)$(W, X) is constructed by pasting together copies ofX, one for each element ofW. The purpose of this chapter is to give the details of this construction. More generally, in 5.1 we describe the construction for an arbitrary groupGtogether with a family of subgroups. This greater generality will not be needed until Chapter 18 when it will be used in the discussion of geometric realizations of buildings.

A...

9. Chapter Six GEOMETRIC REFLECTION GROUPS
(pp. 72-122)

This chapter deals with the classical theory of reflection groups on the three geometries of constant curvature: then-sphere, Euclideann-space, and hyperbolicn-space. Let$\mathbb{X}^{n}$stand for one of these. The main result, Theorem 6.4.3, states that, ifPnis a convex polytope in$\mathbb{X}^{n}$with all dihedral angles integral submultiples ofπ, then the groupWgenerated by the isometric reflections across the codimension one faces ofPnis (1) a Coxeter group and (2) a discrete subgroup of isometries of$\mathbb{X}^{n}$; moreover,$\mathbb{X}^{n}\cong\mathcal{U}(W, P^{n})$. In the spherical case,Pnis a spherical simplex andWis...

10. Chapter Seven THE COMPLEX Σ
(pp. 123-135)

As we mentioned in 1.1, associated to any Coxeter system (W, S) there is a cell complex Σ equipped with a proper, cocompactW-action. It is the natural geometric object associated to (W, S). In this chapter we define Σ and describe some of its basic properties. In Section 8.2 of the next chapter we will prove that Σ is contractible and in 12.3, that its natural piecewise Euclidean metric is CAT(0).

Several other important notions are explained in this chapter. In 7.1 we define the nerveLof (W, S): it is a certain simplicial complex with vertex set...

11. Chapter Eight THE ALGEBRAIC TOPOLOGY OF $\mathcal{U}$ AND OF Σ
(pp. 136-165)

As in Chapter 5, (W, S) is a Coxeter system,Xis a mirrored space overS, and$\mathcal{U} (=\mathcal{U}(W, X))$is the result of applying the basic construction to these data. The two main results of this chapter are formulas for the homology of$\mathcal{U}$(Theorem 8.1.2) and for its cohomology with compact supports (Theorem 8.3.1). The two formulas are similar in appearance and in their proofs. (Of course, they give completely different answers wheneverWis infinite.) We will give alternative proofs in Chapter 15.

The most important applications of these formulas are to the case where$\mathcal{U}$is the...

12. Chapter Nine THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL GROUP AT INFINITY
(pp. 166-175)

Section 9.1 deals with$\pi_{1}(\mathcal{U})$. The universal cover of$\mathcal{U}$can be constructed as a space of the form$\tilde{\mathcal{U}}=\mathcal{U}(\tilde{W}, \tilde{X})$where$\tilde{X}$is the universal cover of the fundamental chamberXand the Coxeter group$\tilde{W}$has a fundamental generator for each component of the inverse image in$\tilde{X}$of each mirror ofX. This gives a short exact sequence computing$\pi_{1}(\mathcal{U})$in terms of$\pi_{1}(X)$and$\tilde{W}$(Theorem 9.1.5).

The main result of 9.2 gives a necessary and sufficient condition for Σ to be simply connected at infinity (Theorem 9.2.2). The condition is phrased in terms of the...

13. Chapter Ten ACTIONS ON MANIFOLDS
(pp. 176-211)

The classical examples of geometric reflection groups in Chapter 6 act on simply connected manifolds of constant curvature. This chapter concerns actions of groups generated by reflections on manifolds without geometric assumptions. In several respects such actions resemble the geometric examples. The first result, Theorem 10.1.13, is that the groupWis a Coxeter group and the manifoldMhas the form$M = \mathcal{U}(W, C)$for some fundamental chamberC. This is easiest to understand in the case where theW-action is smooth (or, at least, locally linear). In this caseCresembles a simple polytope in that it is a manifold...

14. Chapter Eleven THE REFLECTION GROUP TRICK
(pp. 212-229)

The term “reflection group trick” refers to a method for converting a finite aspherical CW complex into a closed aspherical manifold which retracts back onto it. The upshot is that closed aspherical manifolds are at least as complicated as finite aspherical complexes. The basic technique underlying the trick is the semidirect product construction of 9.1.

Following [81], in 11.3 the trick is used to produce examples of topological closed aspherical manifolds which are not homotopy equivalent to smooth closed manifolds. Another consequence is discussed in 11.4: if the Borel Conjecture is true for all closed aspherical manifolds, then its relative...

15. Chapter Twelve Σ IS CAT(0): THEOREMS OF GROMOV AND MOUSSONG
(pp. 230-254)

One of the main results in this book is that the natural piecewise Euclidean metric on Σ(W, S) is nonpositively curved. Although this result will be explained here, a general discussion of nonpositively curved spaces and many of the details of our arguments will be postponed until Appendix I. The “natural piecewise Euclidean metric on Σ” means, roughly, that each cell of Σ is identified with a convex polytope in Euclidean space. This is explained in 12.1. The notion of “nonpositive curvature” is defined via comparison triangles. It means that any sufficiently small geodesic triangle in Σ is “thinner” than...

16. Chapter Thirteen RIGIDITY
(pp. 255-275)

The goal of this chapter is to prove Theorem 13.4.1 which asserts that any Coxeter group of type PM is strongly rigid. Roughly, “type PM” means the nerve of (W, S) is an orientable pseudomanifold. “Strongly rigid” means any two fundamental sets of generators forWare conjugate.

We begin with some basic definitions concerning rigidity in 13.1. Then we show how the “graph twisting” technique of [35] can be used to construct examples of nonrigid Coxeter groups. In 13.2 we explain the one-to-one correspondence between the set of spherical parabolic subgroups inWand the set of their fixed...

17. Chapter Fourteen FREE QUOTIENTS AND SURFACE SUBGROUPS
(pp. 276-285)

In [147] Gromov asked if every word hyperbolic group is either virtually cyclic or “large” in the sense defined below. He also asked if every one-ended word hyperbolic group contains a surface group (see [21]). In this chapter these questions are answered for Coxeter groups. In 14.1 we prove a result of [196] (much of which was proved independently in [140]) that any Coxeter group is either virtually abelian or large. In 14.2 we prove the result of [141] that any Coxeter group is either virtually free or else it contains a surface group.

References include [196, 139, 140, 65,...

18. Chapter Fifteen ANOTHER LOOK AT (CO)HOMOLOGY
(pp. 286-305)

Section 15.1 is separate from the rest of this chapter. It deals with the cohomology ofWwith trivial coefficients in either$\mathbb{Q}$(the rationals) or$\mathbb{F}_{2}$(the field with two elements). We show thatWis rationally acyclic. In the right-angled case, we identify$H^{\ast}(W; \mathbb{F}_{2})$with the face ring of its nerve.

In the remaining sections we continue the line begun in Chapter 8. We answer a basic question left open there: what is theW-module structure on$H_{\ast}(\mathcal{U})$and$H_{c}^{\ast}(\mathcal{U})$? The answer has three components:

an identification of the homology (resp. compactly supported cohomology) of$\mathcal{U}$with...

19. Chapter Sixteen THE EULER CHARACTERISTIC
(pp. 306-314)

Algebraic topology began with Euler’s result that for any cellulation of the 2-sphere, the alternating sum of the number of cells is equal to 2. This leads to the notion of the “Euler characteristic,” defined in 16.1. Given a group of typeVF, we define, in the same section, a rational number called its “Euler characteristic” and give two explicit formulas for it in the case of a Coxeter group. In 16.2 we discuss a conjecture about the Euler characteristic of any even dimensional, closed, aspherical manifoldM^{2k}. It asserts that$(-1)^{k}\chi (M^{2k})\geqslant 0$. The special case for a torsion-free...

20. Chapter Seventeen GROWTH SERIES
(pp. 315-327)

SupposeSa finite set of generators for a groupG. As in 2.1,$l:G\rightarrow \mathbb{N}$is word length. Define a power seriesf(t) (thegrowth seriesofG) by$f(t):=\sum_{g\in G}t^{l(g)}$. Thus,$f(t)=\sum_{n=0}^{\infty}a_{n}t^{n}$, where$a_{n}$is the number of vertices in a sphere of radiusnin Cay(G, S). IfGis finite,f(t) is a polynomial. Under favorable circumstances (for example, whenGis an “automatic group”),f(t) is a rational function. One of the first results in this line was the proof of the rationality of growth series of Coxeter groups. We give the argument in Corollary...

21. Chapter Eighteen BUILDINGS
(pp. 328-343)

Buildings were introduced by Tits as an abstraction of certain incidence geometries which had been studied earlier in the context of algebraic groups (often over finite fields). As a combinatorial object, a building is a generalization of a Coxeter system. It is first of all a “chamber system” as defined below. The additional data for a chamber system to be a building consists of a Coxeter system (W, S) and a “W-valued distance function” on the set of chambers. (This is different from, but equivalent to, the classical definition of a building in [43] as a certain type simplicial complex...

22. Chapter Nineteen HECKE–VON NEUMANN ALGEBRAS
(pp. 344-358)

In 19.1 we define the Hecke algebra associated to a Coxeter system (W, S) and a certain tuple of numbers q. It is a deformation of the group algebra$\mathbb{R}W$and is isomorphic to$\mathbb{R}W$. Hecke algebras arise in the study of buildings of type (W, S) with thickness vector q. Given q, anI-tuple of positive real numbers, in 19.2 we define a nonstandard inner product on$\mathbb{R}W$and then complete it to a Hilbert space$L_{\roman{q}}^{2}(W)$(W). The Hecke algebra$\mathbb{R}_{\roman{q}}W$is a$\ast$-algebra of operators on$L_{\roman{q}}^{2}(W)$and it can be completed to a von...

23. Chapter Twenty WEIGHTED L²-(CO)HOMOLOGY
(pp. 359-400)

Suppose Γ is a discrete group acting properly and cellularly on a CW complexY.$C_{\ast}(Y;\mathbb{R})$and$C^{\ast}(Y;\mathbb{R})$denote the vector spaces of real-valued cellular chains and cochains, respectively.$C^{i}(Y;\mathbb{R})$is the set of all functions on the set ofi-cells ofYwhile$C_{i}(Y;\mathbb{R})$consists of the finitely supported ones. Let$L^{2}C^{i}(Y)\subset C^{i}(Y;\mathbb{R})$be the subspace of square summable cochains.$(L^{2}C^{i}(Y)(=L^{2}C_{i}(Y))$is the Hilbert space completion of$C_{i}(Y;\mathbb{R})$with respect to a standard inner product.) Taking (co)homology, we get theL²-(co)homology spaces ofY. If we are careful to use only closed subspaces in these Hilbert spaces (by taking closures...

24. Appendix A CELL COMPLEXES
(pp. 401-420)
25. Appendix B REGULAR POLYTOPES
(pp. 421-432)
26. Appendix C THE CLASSIFICATION OF SPHERICAL AND EUCLIDEAN COXETER GROUPS
(pp. 433-438)
27. Appendix D THE GEOMETRIC REPRESENTATION
(pp. 439-448)
28. Appendix E COMPLEXES OF GROUPS
(pp. 449-464)
29. Appendix F HOMOLOGY AND COHOMOLOGY OF GROUPS
(pp. 465-476)
30. Appendix G ALGEBRAIC TOPOLOGY AT INFINITY
(pp. 477-486)
31. Appendix H THE NOVIKOV AND BOREL CONJECTURES
(pp. 487-498)
32. Appendix I NONPOSITIVE CURVATURE
(pp. 499-530)
33. Appendix J L²-(CO)HOMOLOGY
(pp. 531-554)
34. Bibliography
(pp. 555-572)
35. Index
(pp. 573-584)