# On hyperbolicity and virtual freeness of automorphism groups

## TL;DR

This paper investigates how the hyperbolic nature and virtual freeness of automorphism groups of graph products and Coxeter groups depend on the structure of their defining graphs, providing characterizations based on graph shape.

## Contribution

It offers new criteria linking the shape of the defining graph to the hyperbolicity and virtual freeness of the automorphism groups of graph products and Coxeter groups.

## Key findings

- Hyperbolicity of automorphism groups depends on the graph's shape.
- Characterization of when automorphism groups are virtually free.
- Structural conditions on graphs determine group properties.

## Abstract

We show that word hyperbolicity of automorphism groups of graph products $G_\Gamma$ and of Coxeter groups $W_\Gamma$ depends strongly on the shape of the defining graph $\Gamma$. We also characterized those $Aut(G_\Gamma)$ and $Aut(W_\Gamma)$ in terms of $\Gamma$ that are virtually free.