Residual properties of automorphism groups of (relatively) hyperbolic groups

Abstract

We show that $Out\left(G\right)$ is residually finite if $G$ is one-ended and hyperbolic relative to virtually polycyclic subgroups. More generally, if $G$ is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer automorphisms preserving the peripheral structure is residually finite. We also show that $Out\left(G\right)$ is virtually residually $p$–finite for every prime $p$ if $G$ is one-ended and toral relatively hyperbolic, or infinitely-ended and virtually residually $p$–finite.