The automorphism group of accessible groups

TL;DR

This paper analyzes the structure of the outer automorphism group of accessible groups, providing a detailed description and criteria for finiteness and finite presentation, with implications for hyperbolic groups.

Contribution

It offers a new framework for understanding Out(G) of accessible groups via extensions of automorphism groups, Dehn twists, and free product automorphisms.

Findings

Out(G) described as extensions of automorphism groups and Dehn twists

Criteria established for Out(G) to be finitely presented

Necessary and sufficient conditions for Out(G) to be finite

Abstract

In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by taking extensions of relative automorphism groups of vertex groups, groups of Dehn twists and groups of automorphisms of free products. We apply this description and obtain a criterion for Out(G) to be finitely presented, as well as a necessary and sufficient condition for Out(G) to be finite. Consequences for hyperbolic groups are discussed.