Cohomology and Euler Characteristics of Coxeter Groups

Coxeter groups are familiar objects in many branches of mathematics. The connections with semisimple Lie theory have been a major motivation for the study of Coxeter groups. (Crystallographic) Coxeter groups are involved in Kac-Moody Lie algebras, which generalize the entire theory of semisimple Lie algebras. Coxeter groups of nite order are known to be nite re ection groups, which appear in invariant theory. Coxeter groups also arise as the transformation groups generated by re ections on manifolds (in a suitable sense). Finally, Coxeter groups are classical objects in combinatorial group theory. In this paper, we discuss the cohomology and the Euler characteristics of ( nitely generated) Coxeter groups. Our emphasis is on the rôle of the parabolic subgroups of nite order in both the Euler characteristics and the cohomology of Coxeter groups. The Euler characteristic is de ned for groups satisfying a suitable cohomological niteness condition. The de nition is motivated by topology, but it has applications to group theory as well. The study of Euler characteristics of Coxeter groups was initiated by J.-P. Serre [22], who obtained the formulae for the Euler characteristics of Coxeter groups, as well as the relation between the Euler characteristics and the Poincar e series of Coxeter groups. The formulae for the Euler characteristics of Coxeter groups were simpli ed by I. M. Chiswell [7]. From his result, one knows that the Euler characteristics of Coxeter groups can be computed in terms of the orders of parabolic subgroups of nite order. On the other hand, for a Coxeter group W , the family of parabolic subgroups of nite order forms a nite simplicial complex F(W ). In general, given a simplicial complex K, the Euler characteristics of Coxeter groups W with F(W ) = K are bounded, but are not unique. However, it follows from the result of M. W. Davis that e(W ) = 0 if F(W ) is a generalized homology 2n-sphere (Theorem 4). Inspired by this result, the author investigated the relation between the Euler characteristics of Coxeter groups W and the simplicial complexes F(W ), and obtained the following results: 1. If F(W ) is a PL-triangulation of some closed 2n-manifold M , then