Volume 14, issue 3 (2014)

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Full-featured peak reduction in right-angled Artin groups

Matthew B Day

Algebraic & Geometric Topology 14 (2014) 1677–1743
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.

Keywords
Whitehead algorithm, peak reduction, automorphism groups of groups, right-angled Artin groups, raags, Hermite normal form
Mathematical Subject Classification 2010
Primary: 20F36
Secondary: 20F28, 15A36