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We prove that the outer automorphism group $Out(G)$ is residually finite whenthe 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 ofacylindrically hyperbolic groups. An endomorphism $\phi$ of a group $G$ is saidto 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 hyperbolicgroup $G$, we show that any commensurating endomorphism of $G$ is inner moduloa small perturbation. This generalizes a theorem of Minasyan and Osin, whichprovided a similar statement in the case when $G$ is relatively hyperbolic. Wethen use this result to study pointwise inner and normal endomorphisms ofacylindrically hyperbolic groups.

Commensurating endomorphisms of acylindrically hyperbolic groups and applications