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Negative curvature in automorphism groups of one-ended hyperbolic groups. (English) Zbl 1515.20219

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 \(\operatorname{Aut}(G)\) of a one-ended hyperbolic group \(G\) turns out to be acylindrically hyperbolic. As a consequence, given a group \(H\) and a morphism \(\varphi : H \to \operatorname{Aut}(G)\), we deduce that the semidirect product \(G \rtimes_\varphi H\) is acylindrically hyperbolic if and only if \(\operatorname{ker}(H \xrightarrow{\varphi} \operatorname{Aut}(G) \to \operatorname{Out}(G))\) is finite.

MSC:

20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
20F28 Automorphism groups of groups
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