Commensurating Endomorphisms of Acylindrically Hyperbolic Groups and Applications
نویسندگان
چکیده
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 φ of a group G is said to be commensurating, if for every g ∈ G some non-zero power of φ(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.
منابع مشابه
Acylindrically hyperbolic groups
We say that a group G is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic groups coincides with many other classes studied in the literature, e.g., the class Cgeom introduced by Hamenstädt, the class of groups admitting a non-elementary weakly properly discontinuous action on a hyperbolic spac...
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