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Full-featured peak reduction in right-angled Artin groups

Matthew B Day

Algebraic & Geometric Topology 14 (2014) 1677–1743

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 AΓ on the set of k–tuples of conjugacy classes from AΓ: 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.

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
Received: 5 April 2013
Revised: 19 November 2013
Accepted: 20 November 2013
Published: 29 May 2014
Matthew B Day
Department of Mathematical Sciences
University of Arkansas
SCEN 301
Fayetteville, AR 72701
United States