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

Allcock, Daniel

Algebr. Geom. Topol. 12, No. 2, 1137-1143 (2012).

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.

MSC:

20F55	Reflection and Coxeter groups (group-theoretic aspects)
20E07	Subgroup theorems; subgroup growth

Keywords:

Coxeter groups; parabolic subgroups; normalizers; nonreflection parts; virtual cohomological dimension

Citations:

Zbl 0935.20027; Zbl 0852.20033

References:

[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