### 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 language | English |
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Pages (from-to) | 1149-1210 |

Number of pages | 62 |

Journal | Groups, Geometry, and Dynamics |

Volume | 10 |

Issue number | 4 |

DOIs | |

Publication status | Published - 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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## Cite this

*Groups, Geometry, and Dynamics*,

*10*(4), 1149-1210. https://doi.org/10.4171/GGD/379