Centralisers of Dehn twist automorphisms of free groups

TL;DR

This paper refines the understanding of centralisers of Dehn twists in free groups, providing an algorithm for their presentations and implications for automorphism actions on CAT(0) spaces.

Contribution

It offers a new algorithm to compute presentations of centralisers and applies this to Nielsen automorphisms, enhancing understanding of automorphism group actions.

Findings

Centralisers have finite index subgroups with finite classifying spaces.

An explicit presentation for the centraliser of Nielsen automorphisms is provided.

Restrictions on Aut(F_n) actions on CAT(0) spaces are derived.

Abstract

We refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(F_n). This gives restrictions to actions of Aut(F_n) on CAT(0) spaces.