Negative curvature in automorphism groups of one-ended hyperbolic groups

Anthony Genevois

Published 2018-10-24Version 1

In this article, we show that some negative curvature may survive when taking the automorphism group of a finitely generated group. More precisely, we prove that the automorphism group $\mathrm{Aut}(G)$ of a one-ended hyperbolic group $G$ which is not virtually a surface group turns out to be acylindrically hyperbolic. As a consequence, given an automorphism $\varphi \in \mathrm{Aut}(G)$, we deduce that the semidirect product $G \rtimes_\varphi \mathbb{Z}$ is acylindrically hyperbolic if and only if $\varphi$ has infinite order in $\mathrm{Out}(G)$.