Full-featured peak reduction in right-angled Artin groups

TL;DR

This paper extends peak-reduction techniques to right-angled Artin groups, enabling decidability of orbit membership and finite presentability of stabilizers, with practical procedures for these computations.

Contribution

It introduces a new peak-reduction theorem for automorphisms of right-angled Artin groups and applies it to solve orbit and stabilizer problems.

Findings

Orbit membership is decidable.

Stabilizers are finitely presentable.

Procedures for checking orbit membership and stabilizer presentations are provided.

Abstract

We prove a new version of the classical peak-reduction theorem for automorphisms of free groups in the setting of right-angled Artin groups. We use this peak-reduction theorem to prove two important corollaries about the action of the automorphism group of a right-angled Artin group $A_\Gamma$ on the set of $k$-tuples of conjugacy classes from $A_\Gamma$: orbit membership is decidable, and stabilizers are finitely presentable. Further, we explain procedures for checking orbit membership and building presentations of stabilizers. This improves on a previous result of the author's. We overcome a technical difficulty from the previous work by considering infinite generating sets for the automorphism groups. The method also involves a variation on the Hermite normal form for matrices.