Vol. 314, No. 1, 2021

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Outer automorphism groups of graph products: subgroups and quotients

Andrew Sale and Tim Susse

Vol. 314 (2021), No. 1, 161–208
Abstract

We show that the outer automorphism groups of graph products of finitely generated abelian groups satisfy the Tits alternative, are residually finite, their so-called Torelli subgroups are finitely generated, and they satisfy a dichotomy between being virtually nilpotent and containing a nonabelian free subgroup that is determined by a graphical condition on the underlying labeled graph.

Graph products of finitely generated abelian groups simultaneously generalize right-angled Artin groups (RAAGs) and right-angled Coxeter groups (RACGs), providing a common framework for studying these groups. Our results extend a number of known results for the outer automorphism groups of RAAGs and/or RACGs by a variety of authors, including Caprace, Charney, Day, Ferov, Guirardel, Horbez, Minasyan, Vogtmann, Wade, and the current authors.

Keywords
graph product, automorphism group, Tits alternative, residually finite
Mathematical Subject Classification
Primary: 20F28, 20F65
Milestones
Received: 5 September 2019
Revised: 10 June 2021
Accepted: 10 June 2021
Published: 15 October 2021
Authors
Andrew Sale
University of Hawaii at Manoa
Honoulu, HI
United States
Tim Susse
Bard College at Simon’s Rock
Great Barrington, MA
United States