Residual properties of automorphism groups of (relatively) hyperbolic groups

TL;DR

This paper proves that the outer automorphism groups of certain relatively hyperbolic groups are residually finite or virtually p-residually finite, extending understanding of their algebraic properties and automorphism structures.

Contribution

It establishes residual finiteness and virtual p-residual finiteness of Out(G) for various classes of relatively hyperbolic groups, generalizing previous results.

Findings

Out(G) is residually finite for one-ended hyperbolic relative to virtually polycyclic subgroups.

Out(G) is residually finite when G is hyperbolic relative to proper residually finite subgroups.

Out(G) is virtually p-residually finite for toral relatively hyperbolic groups or infinitely-ended virtually p-residually finite groups.

Abstract

We show that Out(G) is residually finite if G is a one-ended group that is 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(G) is virtually p-residually finite for every prime p if G is one-ended and toral relatively hyperbolic, or infinitely-ended and virtually p-residually finite.