Commensurating endomorphisms of acylindrically hyperbolic groups and applications

Yago Antolín, Ashot Minasyan, Alessandro Sisto

Research output: Contribution to journalReview article

10 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1149-1210
Number of pages62
JournalGroups, Geometry, and Dynamics
Volume10
Issue number4
DOIs
Publication statusPublished - 2016

Keywords

  • 3-Manifold groups
  • Acylindrically hyperbolic groups
  • Commensurating endomorphisms
  • Hyperbolically embedded subgroups
  • Outer automorphism groups
  • Pointwise inner automorphisms
  • Right angled Artin groups

ASJC Scopus subject areas

  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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