The Tits alternative for the automorphism group of a free product

TL;DR

This paper proves that the outer automorphism group of a free product satisfies the Tits alternative under certain conditions, extending the property to complex group constructions like right-angled Artin groups.

Contribution

It establishes the Tits alternative for outer automorphism groups of free products with specific indecomposable factors, including applications to right-angled Artin groups.

Findings

Outer automorphism groups of free products satisfy the Tits alternative.

The result applies to right-angled Artin groups.

The Tits alternative holds for certain relatively hyperbolic groups.

Abstract

Let $G=G_1\ast\dots\ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all $i\in\{1,\dots,k\}$, both $G_i$ and its outer automorphism group $\text{Out}(G_i)$ satisfy the Tits alternative, then $\text{Out}(G)$ satisfies the Tits alternative. As an application, we prove that the Tits alternative holds for outer automorphism groups of right-angled Artin groups, and of torsion-free groups that are hyperbolic relative to a finite family of virtually polycyclic groups.