Full-featured peak reduction in right-angled Artin groups

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. 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.