Commensurating endomorphisms of acylindrically hyperbolic groups and applications Article Swipe
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· 2017
· Open Access
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· DOI: https://doi.org/10.4171/ggd/379
· OA: W1928123342
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.
