Negative curvature in automorphism groups of one-ended hyperbolic groups

Abstract

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 $G$ turns out to be acylindrically hyperbolic. As a consequence, given a group $H$ and a morphism $\phi :H\to \mathrm{Aut}(G)$, we deduce that the semidirect product $G{⋊}_{\phi }H$ is acylindrically hyperbolic if and only if $\ker (H\stackrel{\phi }{\to }\mathrm{Aut}(G)\to \mathrm{Out}(G))$ is finite.