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

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Cited by in corpus (22)

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• Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups
• McCool groups of toral relatively hyperbolic groups
• Acylindrically hyperbolic groups with exotic properties
• Extending higher dimensional quasi-cocycles
• Formal conjugacy growth in acylindrically hyperbolic groups
• Acylindrical hyperbolicity of groups acting on trees
• Hyperbolic HHS II: Graphs of hierarchically hyperbolic groups
• Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups
• On automorphisms and splittings of special groups
• On conjugacy separability of fibre products
• Farrell-Jones via Dehn fillings
• Random walks and quasi-convexity in acylindrically hyperbolic groups
• Acylindrical hyperbolicity of cubical small-cancellation groups
• Wreath-like products of groups and their von Neumann algebras I: $W^{\ast }$-superrigidity
• Formal solutions and the first-order theory of acylindrically hyperbolic groups
• Rigidity of mapping class groups mod powers of twists
• Flat braid groups, right-angled Artin groups, and commensurability
• Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces
• On subgroups of right angled Artin groups with few generators