Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups

Abstract

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 ${\mathbb{F}}_n{⋊}_{\phi }\mathbb{Z}$ is acylindrically hyperbolic if and only if $\phi$ has infinite order in $\mathrm{Out}({\mathbb{F}}_n)$.

arXiv comment31 pages. To appear in Journal of Topology

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Cited by in corpus (2)

• Aut-invariant quasimorphisms on groups
• Train Tracks on Graphs of Groups and Outer Automorphisms of Hyperbolic Groups