Automorphisms of graph products of groups and acylindrical hyperbolicity

TL;DR

This paper investigates the acylindrical hyperbolicity of automorphism groups of graph products of groups, establishing new results that extend to right-angled Artin and Coxeter groups, with implications for their algebraic properties.

Contribution

It proves that automorphism groups of certain graph products are acylindrically hyperbolic or infinite dihedral, providing new insights into their structure and properties.

Findings

Automorphism groups are often acylindrically hyperbolic or infinite dihedral.

Automorphism groups of most graph products are SQ-universal.

Many automorphism groups do not satisfy Kazhdan's property (T).

Abstract

This article is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. Our main result is that, if $\Gamma$ is a finite graph which contains at least two vertices and is not a join and if $\mathcal{G}$ is a collection of finitely generated irreducible groups, then either $\Gamma \mathcal{G}$ is infinite dihedral or $\mathrm{Aut}(\Gamma \mathcal{G})$ is acylindrically hyperbolic. This theorem is new even for right-angled Artin groups and right-angled Coxeter groups. Various consequences are deduced from this statement and from the techniques used to prove it. For instance, we show that the automorphism groups of most graph products verify vastness properties such as being SQ-universal; we show that many automorphism groups of graph products do not satisfy Kazhdan's property (T); we solve the isomorphism problem between graph products in some cases; and we show that a graph product of coarse median groups, as defined by Bowditch, is coarse median itself.