Negative curvature in automorphism groups of one-ended hyperbolic groups (Q2335811)

Summary: 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 \(\varphi : H \to \mathrm{Aut}(G)\), we deduce that the semidirect product \(G \rtimes_\varphi H\) is acylindrically hyperbolic if and only if \(\mathrm{ker}(H \xrightarrow{\varphi} \mathrm{Aut}(G) \to \mathrm{Out}(G))\) is finite.