Non-residually finite groups hyperbolic relative to residually finite subgroups

2019/03/03 by Jan Kim, Kim, Jan, Donghi Lee +1
Computer Science · Mathematics · #20F65 #20F67 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1903.00838

openalex publication_date 2019/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let m and k be integers such that |m|, |k| >1 and gcd (m,k)=1. We show that all Baumslag-Solitar groups BS(m,mk) are non-residually finite groups hyperbolic relative to residually finite subgroups. By a result of Osin (2007), this implies that there exists a non-residually finite hyperbolic group, thus solving a long-standing open problem of Gromov (1987). We also show that all BS(m, mk) are non-Hopfian groups hyperbolic relative to Hopfian subgroups.

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