We prove that the outer automorphism group Out is residually finite when the group is virtually compact special (in the sense of Haglund and Wise) or when 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 of a group is said to be commensurating, if for every some non-zero power of is conjugate to a non-zero power of . Given an acylindrically hyperbolic group , we show that any commensurating endomorphism of is inner modulo a small perturbation. This generalizes a theorem of Minasyan and Osin, which provided a similar statement in the case when is relatively hyperbolic. We then use this result to study pointwise inner and normal endomorphisms of acylindrically hyperbolic groups.
Cite this article
Yago Antolín, Ashot Minasyan, Alessandro Sisto, Commensurating endomorphisms of acylindrically hyperbolic groups and applications. Groups Geom. Dyn. 10 (2016), no. 4, pp. 1149–1210DOI 10.4171/GGD/379