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 or is either free or has a free subgroup of index .
"Normalizers of parabolic subgroups of Coxeter groups." Algebr. Geom. Topol. 12 (2) 1137 - 1143, 2012. https://doi.org/10.2140/agt.2012.12.1137