Algebraic & Geometric Topology

Normalizers of parabolic subgroups of Coxeter groups

Daniel Allcock

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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
Received: 13 September 2011
Accepted: 18 January 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document
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
Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]

Keywords
Coxeter group parabolic subgroup nonreflection part

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


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References

  • D Allcock, Reflection centralizers in Coxeter groups, in preparation
  • R E Borcherds, Coxeter groups, Lorentzian lattices, and $K3$ surfaces, Internat. Math. Res. Notices (1998) 1011–1031
  • 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)