Geodesic
Normalizers of parabolic subgroups of Coxeter groups
Allcock, Daniel1
1Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin TX 78712, USA
Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 1137-1143
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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.

DOI : 10.2140/agt.2012.12.1137
Classification : 20F55
Keywords: Coxeter group, parabolic subgroup, nonreflection part

Allcock, Daniel  1

1 Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin TX 78712, USA 
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Allcock, Daniel. Normalizers of parabolic subgroups of Coxeter groups. Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 1137-1143. doi: 10.2140/agt.2012.12.1137

[1] D Allcock, Reflection centralizers in Coxeter groups, in preparation

[2] R E Borcherds, Coxeter groups, Lorentzian lattices, and K3 surfaces, Internat. Math. Res. Notices (1998) 1011

[3] B Brink, On centralizers of reflections in Coxeter groups, Bull. London Math. Soc. 28 (1996) 465

[4] B Brink, R B Howlett, Normalizers of parabolic subgroups in Coxeter groups, Invent. Math. 136 (1999) 323

[5] J E Humphreys, Reflection groups and Coxeter groups, 29, Cambridge Univ. Press (1990)

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