Aperiodicity properties of automorphism groups of free products

TL;DR

This paper investigates the automorphism groups of free products of finitely presented groups, establishing their aperiodicity properties and torsion-freeness under certain conditions, with applications to hyperbolic groups and free groups.

Contribution

It proves that certain subgroups of automorphism groups have aperiodicity properties, are torsion free, and fixed points for periodic conjugacy classes, extending to hyperbolic groups.

Findings

The subgroup (G,) is torsion free.

Periodic conjugacy classes are fixed by automorphisms in (G,).

Results apply to hyperbolic groups and provide alternative proofs for known theorems.

Abstract

Let $G=G_1 \ast \ldots \ast G_k \ast F_N$ be a free product of finitely presented groups, where $F_N$ is a free group of rank $N \in \mathbb{N}$. Let $\mathrm{Out}(G,\mathcal{G})$ be the subgroup of $\mathrm{Out}(G)$ preserving the set of conjugacy classes $\mathcal{G}=\{[G_1],\ldots,[G_k]\}$. Under natural conditions on the groups $G_i$ with $i \in \{1,\ldots,k\}$, we prove that the group $\mathrm{Out}(G,\mathcal{G})$ has a finite index subgroup $\mathrm{IA}(G,\mathcal{G},3)$ with notable aperiodicity properties. We show that the group $\mathrm{IA}(G,\mathcal{G},3)$ is torsion free and, if $\phi \in \mathrm{IA}(G,\mathcal{G},3)$, every $\phi$-periodic conjugacy class of elements of $G$ is in fact fixed by $\phi$ and every $\phi$-periodic conjugacy class of free factors of $G$ is fixed by $\phi$. As an application, we prove that, for every toral relatively hyperbolic group $G$, the group $\mathrm{Out}(G)$ has a finite index subgroup $\mathrm{IA}(G,3)$ with the same above mentioned aperiodicity properties. We in particular give another proof of the theorem, due to Handel-Mosher, that the kernel of the action of $\mathrm{Out}(F_N)$ on $H_1(F_N,\mathbb{Z}/3\mathbb{Z})$ satisfies natural aperiodicity properties.