Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups

Anthony Genevois, Camille Horbez

Published 2020-02-04Version 1

We prove that the automorphism group of every infinitely-ended finitely generated group is acylindrically hyperbolic. In particular $\mathrm{Aut}(\mathbb{F}_n)$ is acylindrically hyperbolic for every $n\ge 2$. More generally, if $G$ is a group which is not virtually cyclic, and hyperbolic relative to a finite collection $\mathcal{P}$ of finitely generated proper subgroups, then $\mathrm{Aut}(G,\mathcal{P})$ is acylindrically hyperbolic. As a consequence, a free-by-cyclic group $\mathbb{F}_n\rtimes_{\varphi}\mathbb{Z}$ is acylindrically hyperbolic if and only if $\varphi$ has infinite order in $\mathrm{Out}(\mathbb{F}_n)$.