## Algebraic & Geometric Topology

### Normalizers of parabolic subgroups of Coxeter groups

Daniel Allcock

#### 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 $D5$ or $Amodd$ is either free or has a free subgroup of index $2$.

#### Article information

Source
Algebr. Geom. Topol., Volume 12, Number 2 (2012), 1137-1143.

Dates
Accepted: 18 January 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715383

Digital Object Identifier
doi:10.2140/agt.2012.12.1137

Mathematical Reviews number (MathSciNet)
MR2928907

Zentralblatt MATH identifier
1248.20045

Subjects

#### Citation

Allcock, Daniel. Normalizers of parabolic subgroups of Coxeter groups. Algebr. Geom. Topol. 12 (2012), no. 2, 1137--1143. doi:10.2140/agt.2012.12.1137. https://projecteuclid.org/euclid.agt/1513715383

#### References

• D Allcock, Reflection centralizers in Coxeter groups, in preparation
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• B Brink, On centralizers of reflections in Coxeter groups, Bull. London Math. Soc. 28 (1996) 465–470
• B Brink, R B Howlett, Normalizers of parabolic subgroups in Coxeter groups, Invent. Math. 136 (1999) 323–351
• J E Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press (1990)