Normalizers of parabolic subgroups of Coxeter groups. (English) Zbl 1248.20045

Summary: We improve a bound of R. E. Borcherds on the virtual cohomological dimension of the nonreflection part of the normalizer of a parabolic subgroup of a Coxeter group [Int. Math. Res. Not. 1998, No. 19, 1011-1031 (1998; Zbl 0935.20027)]. 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 B. Brink’s result that the nonreflection part of a reflection centralizer is free [Bull. Lond. Math. Soc. 28, No. 5, 465-470 (1996; Zbl 0852.20033)]. Namely, the nonreflection part of the normalizer of parabolic subgroup of type \(D_5\) or \(A_{m\text{ odd}}\) is either free or has a free subgroup of index 2.


20F55 Reflection and Coxeter groups (group-theoretic aspects)
20E07 Subgroup theorems; subgroup growth
Full Text: DOI arXiv


[1] D Allcock, Reflection centralizers in Coxeter groups, in preparation · Zbl 1283.20042
[2] R E Borcherds, Coxeter groups, Lorentzian lattices, and \(K3\) surfaces, Internat. Math. Res. Notices (1998) 1011 · Zbl 0935.20027
[3] B Brink, On centralizers of reflections in Coxeter groups, Bull. London Math. Soc. 28 (1996) 465 · Zbl 0852.20033
[4] B Brink, R B Howlett, Normalizers of parabolic subgroups in Coxeter groups, Invent. Math. 136 (1999) 323 · Zbl 0926.20024
[5] J E Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press (1990) · Zbl 0725.20028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.