# Outer automorphism groups of graph products: subgroups and quotients

## TL;DR

This paper investigates the outer automorphism groups of graph products of finitely generated abelian groups, demonstrating they satisfy the Tits alternative, are residually finite, and exhibit a dichotomy based on a graphical condition.

## Contribution

It extends known results on automorphism groups of RAAGs and RACGs to a broader class of graph products, establishing new properties and conditions.

## Key findings

- Outer automorphism groups satisfy the Tits alternative.
- They are residually finite.
- Torelli subgroups are finitely generated.

## 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 non-abelian free subgroup that is determined by a graphical condition on the underlying labelled graph. Graph products of finitely generated abelian groups simultaneously generalize right-angled Artin groups (RAAGs) and right-angled Coxter 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.