A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection

Details A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection

2006-03-28

arxiv.org/abs/math/0603667v2

Mathematics

Group Theory

21 pages; previously entitled "On reflections in Coxeter groups commuting with a reflection" (v2) presentation improved

Scientific paper

Given a reflection $r$ in a Coxeter group $W$ (possibly of infinite rank), we consider the subgroup of $W$ generated by the reflections in $W$ having (-1)-eigenvectors orthogonal to the (-1)-eigenvector of $r$. In this paper, we determine completely when this subgroup is finitely generated, by using preceding results on centralizers of reflections. This result provides a new and naturally constructed example of a family of non-finitely generated subgroups of finitely generated groups.

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