Abstract. We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNNextensions, one-relator groups, automorphism groups of polynomial algebras, 3-manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain some results about the algebraic structure, analytic properties and measure equivalence rigidity of groups from these classes.

We prove that finite index subgroups of right angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that Bestvina-Brady groups are conjugacy separable and have solvable conjugacy problem.

In this paper we study residual properties of relatively hyperbolic groups. In particular, we show that if a group G is non-elementary and hyperbolic relative to a collection of proper subgroups, then G is SQuniversal.

Abstract. An automorphism α of a group G is normal if it fixes every normal subgroup of G setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we show that for any such group G, Inn(G) has finite index in the subgroup Aut n (G) of normal automorphisms. If, in addition, G is non-elementary and has no finite non-trivial normal subgroups, then Aut n (G) = Inn (G). As an application, we show that Out(G) is residually finite for every finitely generated residually finite group G with infinitely many ends.

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.