Normal automorphisms of relatively hyperbolic groups

TL;DR

This paper characterizes normal automorphisms of relatively hyperbolic groups, showing they are closely related to inner automorphisms, and applies these results to demonstrate residual finiteness of outer automorphism groups for certain groups.

Contribution

It provides an algebraic description of normal automorphisms in relatively hyperbolic groups and establishes their relation to inner automorphisms, extending understanding of automorphism structures.

Findings

Inner automorphisms have finite index in normal automorphisms

Normal automorphisms coincide with inner automorphisms under certain conditions

Outer automorphism groups are residually finite for specific finitely generated groups

Abstract

An automorphism $\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group $G$, $Inn(G)$ has finite index in the subgroup $Aut_n(G)$ of normal automorphisms. If, in addition, $G$ is non-elementary and has no non-trivial finite normal subgroups, then $Aut_n(G)=Inn(G)$. As an application, we show that $Out(G)$ is residually finite for every finitely generated residually finite group $G$ with more than one end.