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

2006/03/28 by Koji Nuida, Nuida, Koji
Computer Science · Mathematics · #20E34 #20F55 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0603667

openalex publication_date 2006/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Centralizers of reflections and reflection-independence of Coxeter groups
• On reflection subgroups of finite Coxeter groups
• Reflection centralizers in Coxeter groups
• Reflections in abstract Coxeter groups
• On non-conjugate Coxeter elements in well-generated reflection groups