Non-residually finite groups hyperbolic relative to residually finite subgroups

Jan Kim; Donghi Lee

Non-residually finite groups hyperbolic relative to residually finite subgroups

Group Theory 2019-03-07 v2

Authors: Jan Kim , Donghi Lee

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.

Keywords

hyperbolic groups group theory hyperbolic geometry

Cite

@article{arxiv.1903.00838,
  title  = {Non-residually finite groups hyperbolic relative to residually finite subgroups},
  author = {Jan Kim and Donghi Lee},
  journal= {arXiv preprint arXiv:1903.00838},
  year   = {2019}
}

Comments

Due to an important error in Section 4, we would like to politely ask to withdraw this manuscript. We greatly appreciate Professors Yves Cornulier, Francois Dahmani, Anthony Genevois, Daniel Groves, Ilya Kapovich, Dennis Osin and Henry Wilton for their kind comments and pointing out the error

Non-residually finite groups hyperbolic relative to residually finite subgroups

Abstract

Keywords

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