Elementary subgroups of relatively hyperbolic groups and bounded generation

2004/04/06 by Denis Osin, D. V. Osin, Osin, D. V. · 2 citations
Computer Science · Mathematics · #20F65 #20F67 #20F69 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #msc:20F65 #msc:20F67 #msc:20F69 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0404118

21 pages

arxiv created 2004/04/06 · openalex publication_date 2004/04/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a group hyperbolic relative to a collection of subgroups \H_λ,λ∈ Λ\ . We say that a subgroup Q≤ G is hyperbolically embedded into G, if G is hyperbolic relative to \H_λ,λ∈ Λ\ ∪ \Q\ . In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element g∈ G has infinite order and is not conjugate to an element of H_λ, λ∈ Λ, then the (unique) maximal elementary subgroup contained g is hyperbolically embedded into G. This allows to prove that if G is boundedly generated, then G is elementary or H_λ=G for some λ∈ Λ.

Cited by

• A non-Hopfian ascending HNN-extension of a finitely presented Hopfian group
• Parabolic cut pairs in boundaries of relatively hyperbolic groups

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