Normalizers of Parabolic Subgroups of Coxeter Groups

TL;DR

This paper refines bounds on the virtual cohomological dimension of certain Coxeter group normalizers, extending known results about their structure and freeness properties based on Coxeter diagram components.

Contribution

It improves Borcherds' bound on the cohomological dimension and extends Brink's results on the freeness of non-reflection parts of normalizers.

Findings

Bound in terms of Coxeter diagram components rather than nodes

Extension of Brink's result to types D5 and Aodd

Non-reflection parts are either free or have a free subgroup of index 2

Abstract

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 free. Namely, the non-reflection part of the normalizer of parabolic subgroup of type D5 or Aodd is either free or has a free subgroup of index 2.