# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.00838