Euler Characteristics of Coxeter Groups, PL-Triangulations of Closed Manifolds, and Cohomology of Subgroups of Artin Groups

Abstract

The motivation for the theory of Euler characteristics of groups, which was introduced by C. T. C. Wall [21], was topology, but it has interesting connections to other branches of mathematics such as group theory and number theory. This paper investigates Euler characteristics of Coxeter groups and their applications. In his paper [20], J.-P. Serre obtained several fundamental results concerning the Euler characteristics of Coxeter groups. In particular, he obtained a recursive formula for the Euler characteristic of a Coxeter group, as well as its relation to the Poincaré series (see §3). Later, I. M. Chiswell obtained in [10] a formula expressing the Euler characteristic of a Coxeter group in terms of orders of finite parabolic subgroups (Theorem 1). These formulae enable us to compute Euler characteristics of arbitrary Coxeter groups. On the other hand, the Euler characteristics of Coxeter groups W happen to be intimately related to their associated complexes FW, which are defined by means of the posets of nontrivial parabolic subgroups of finite order (see §2.1 for the precise definition). In particular, it follows from the recent result of M. W. Davis [13] that if FW is a product of a simplex and a generalized homology 2n-sphere, then the Euler characteristic of W is zero (Corollary 3.1). The first objective of this paper is to generalize the previously mentioned result to the case when FW is a PL-triangulation of a closed 2n-manifold which is not necessarily a homology 2n-sphere. In other words (as given below in Theorem 3), if W is a Coxeter group such that FW is a PL-triangulation of a closed 2n-manifold, then the Euler characteristic of W is equal to 1−χ(FW)/2.