Non-residually finite groups hyperbolic relative to residually finite subgroups

TL;DR

This paper demonstrates that certain Baumslag-Solitar groups are non-residually finite and hyperbolic relative to residually finite subgroups, solving a long-standing open problem in geometric group theory.

Contribution

It proves that all $BS(m,mk)$ groups are non-residually finite hyperbolic relative to residually finite subgroups, providing new examples of hyperbolic groups with specific residual properties.

Findings

All $BS(m,mk)$ are non-residually finite hyperbolic relative to residually finite subgroups.

Existence of a non-residually finite hyperbolic group, answering Gromov's open problem.

All $BS(m,mk)$ are non-Hopfian hyperbolic relative to Hopfian subgroups.

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.