On the Virtual Cohomological Dimensions of Coxeter Groups

Normalizers of Parabolic Subgroups of Coxeter Groups

We improve a bound of Borcherds on the virtual cohomological dimension of the non-reflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter subdiagram rather than the number of nodes. A consequence is an extension of Brink’s result that the non-reflection part of a reflection centralizer is fre...

The Deformation Theory of Discrete Reflection Groups and Projective Structures

We study deformations of discrete groups generated by linear reflections and associated geometric structures on orbifolds via cohomology of Coxeter groups with coefficients in the adjoint representation associated to a discrete representation. We completely describe a cochain complex that computes this cohomology for an arbitrary discrete reflection group and, as a consequence of this descripti...

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 in...

Complete Growth Series of Coxeter Groups with more than Three Generators

Accessing the cohomology of discrete groups above their virtual cohomological dimension

We introduce a method to explicitly determine the Farrell–Tate cohomology of discrete groups. We apply this method to the Coxeter triangle and tetrahedral groups as well as to the Bianchi groups, i.e. PSL2(O) for O the ring of integers in an imaginary quadratic number field, and to their finite index subgroups. We show that the Farrell–Tate cohomology of the Bianchi groups is completely determi...