JournalsggdVol. 10, No. 4pp. 1149–1210

Commensurating endomorphisms of acylindrically hyperbolic groups and applications

  • Yago Antolín

    Vanderbilt University, Nashville, USA
  • Ashot Minasyan

    University of Southampton, UK
  • Alessandro Sisto

    ETH Zürich, Switzerland
Commensurating endomorphisms of acylindrically hyperbolic groups and applications cover
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Abstract

We prove that the outer automorphism group Out(G)(G) is residually finite when the group GG is virtually compact special (in the sense of Haglund and Wise) or when GG 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\phi of a group GG is said to be commensurating, if for every gGg \in G some non-zero power of ϕ(g)\phi(g) is conjugate to a non-zero power of gg. Given an acylindrically hyperbolic group GG, we show that any commensurating endomorphism of GG is inner modulo a small perturbation. This generalizes a theorem of Minasyan and Osin, which provided a similar statement in the case when GG 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. 1149–1210

DOI 10.4171/GGD/379