Groups, Geometry, and Dynamics


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Volume 10, Issue 4, 2016, pp. 1149–1210
DOI: 10.4171/GGD/379

Published online: 2017-01-26

Commensurating endomorphisms of acylindrically hyperbolic groups and applications

Yago Antolín[1], Ashot Minasyan[2] and Alessandro Sisto[3]

(1) Vanderbilt University, Nashville, USA
(2) University of Southampton, UK
(3) ETH Zürich, Switzerland

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 $\phi$ of a group $G$ is said to be commensurating, if for every $g \in G$ some non-zero power of $\phi(g)$ 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.

Keywords: Acylindrically hyperbolic groups, hyperbolically embedded subgroups, commensurating endomorphisms, pointwise inner automorphisms, right angled Artin groups, outer automorphism groups, 3-manifold groups

Antolín Yago, Minasyan Ashot, Sisto Alessandro: Commensurating endomorphisms of acylindrically hyperbolic groups and applications. Groups Geom. Dyn. 10 (2016), 1149-1210. doi: 10.4171/GGD/379