Commensurating endomorphisms of acylindrically hyperbolic groups and applications

Abstract

We prove that the outer automorphism group $Out(G)$ is residually finite when the group $G$ is virtually compact special (in the sense of Haglund and Wise) or when $G$ is isomorphic to the fundamental group of some compact $3$-manifold. To prove these results we characterize commensurating endomorphisms of acylindrically hyperbolic groups. An endomorphism $\varphi$ of a group $G$ is said to be commensurating, if for every $g\in G$ some non-zero power of $\varphi (g)$ is conjugate to a non-zero power of $g$. Given an acylindrically hyperbolic group $G$, we show that any commensurating endomorphism of $G$ is inner modulo a small perturbation. This generalizes a theorem of Minasyan and Osin, which provided a similar statement in the case when $G$ is relatively hyperbolic. We then use this result to study pointwise inner and normal endomorphisms of acylindrically hyperbolic groups.

47 pages. v4: final version, to appear in Groups, Geometry and Dynamics

References in corpus (1)

• On metric relative hyperbolicity

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