Outer automorphism groups of graph products: subgroups and quotients

Andrew Sale and Tim Susse

Vol. 314 (2021), No. 1, 161–208

DOI: 10.2140/pjm.2021.314.161

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