Volume 12, issue 2 (2012)

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Normalizers of parabolic subgroups of Coxeter groups

Daniel Allcock

Algebraic & Geometric Topology 12 (2012) 1137–1143
Abstract

We improve a bound of Borcherds on the virtual cohomological dimension of the nonreflection 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 nonreflection part of a reflection centralizer is free. Namely, the nonreflection part of the normalizer of parabolic subgroup of type ${D}_{5}$ or ${A}_{m\phantom{\rule{0.3em}{0ex}}odd}$ is either free or has a free subgroup of index $2$.

Keywords
Coxeter group, parabolic subgroup, nonreflection part
Primary: 20F55