Peak reduction and finite presentations for automorphism groups of right-angled Artin groups

TL;DR

This paper extends the peak-reduction algorithm to right-angled Artin groups, providing a finite presentation for their automorphism groups and exploring a stronger generalization with counterexamples.

Contribution

It generalizes Whitehead's peak-reduction algorithm and McCool's presentation to a broader class of groups, offering new tools for understanding automorphism groups.

Findings

A generalized peak-reduction algorithm for right-angled Artin groups

A finite presentation for automorphism groups of these groups

Counterexamples and special case results for stronger generalizations

Abstract

We generalize the peak-reduction algorithm (Whitehead's theorem) for free groups to a theorem about a general right-angled Artin group A_Gamma. As an application, we find a finite presentation for the automorphism group Aut A_Gamma that generalizes McCool's presentation for the automorphism group of a finite rank free group. We also give consider a stronger generalization of peak-reduction, giving a counterexample and proving a special case.