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