Non-residually finite groups hyperbolic relative to residually finite subgroups

Let $m$ and $k$ be integers such that $\mathrm{\mid }m\mathrm{\mid },\mathrm{\mid }k\mathrm{\mid }>1$ 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.