The Tits alternative for the automorphism group of a free product

Abstract

Let $G=G_1\ast \cdots \ast G_k\ast F$ 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.

v2: Revised version: introduction rewritten, exposition improved, typos corrected

References in corpus (4)

• Reducing systems for very small trees
• Subgroup classification in Out(F_n)
• Actions of finitely generated groups on R-trees
• A short proof of Handel and Mosher's alternative for subgroups of $\text{Out}(F_N)$

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• Limit trees for free group automorphisms: universality
• Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$
• Train track maps for graphs of groups