Negative curvature in automorphism groups of one-ended hyperbolic groups

TL;DR

This paper demonstrates that the automorphism groups of one-ended hyperbolic groups retain negative curvature properties, specifically being acylindrically hyperbolic, which influences the structure of related semidirect products.

Contribution

It proves that automorphism groups of one-ended hyperbolic groups are acylindrically hyperbolic, revealing persistent negative curvature properties in these automorphism groups.

Findings

Automorphism groups of one-ended hyperbolic groups are acylindrically hyperbolic.

Semidirect products with such groups are acylindrically hyperbolic under certain conditions.

The kernel of the induced map to Out(G) determines the hyperbolicity of the semidirect product.

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