Aperiodicity properties of automorphism groups of free products

Abstract

Let $G=G_1\ast \dots \ast G_k\ast F_N$ 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 $\mathcal{G}=\{[G_1],\dots ,[G_k]\}$. Under natural conditions on the groups $G_i$ with $i\in \{1,\dots ,k\}$, 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 $\varphi \in \mathrm{IA}(G,\mathcal{G},3)$, every $\varphi$-periodic conjugacy class of elements of $G$ is in fact fixed by $\varphi$ and every $\varphi$-periodic conjugacy class of free factors of $G$ is fixed by $\varphi$. 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.

59 pages