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...

We study the action of a relatively hyperbolic group on its boundary, by methods of symbolic dynamics. Under a condition on the parabolic subgroups, we show that this dynamical system is finitely presented. We give examples where this condition is satisfied, including geometrically finite kleinian groups. Associated to any word-hyperbolic group Γ, there is a dynamical system arising from the ac...

We show that if G is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group G, such that G is a subgroup of G but G is not quasiconvex in G.

The purpose of this note is to present a criterion for an infinite collection of distinct hyperbolic 3-manifolds to be commensurably infinite. (Here, a collection of hyperbolic 3-manifolds is commensurably infinite if it contains representatives from infinitely many commensurability classes.) Namely, such a collection M is commensurably infinite if there is a uniform upper bound on the volumes ...

We show that every virtually torsion-free subgroup of the outer au-tomorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite. We also generalise the main result for relatively hyperbolic groups.

We study the subgroup structure of discrete groups which share cohomological properties which resemble non-negative curvature. Examples include all Gromov hyperbolic groups. A subgroup H ⊂ G is called s-normal, if H∩H is infinite for all g ∈ G. We provide strong restrictions on the possible s-normal subgroups of a Gromov hyperbolic group, or more generally a ’negatively curved’ group. Another r...

The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none the automorphic image of another, such that each normally generates the group.

By constructing, in the relative case, objects analoguous to Rips and Sela’s canonical representatives, we prove that the set of conjugacy classes of images by morphisms without accidental parabolic, of a finitely presented group in a relatively hyperbolic group, is finite. An important result of W.Thurston is : Theorem 0.1 ([T] 8.8.6) Let S be any hyperbolic surface of finite area, and N any g...

The notion of a group hyperbolic relative to a collection of subgroups was originally proposed by Gromov [15]. Since then many definitions of relatively hyperbolic groups were suggested by Bowditch [5], Farb [13], Drutu-Osin-Sapir [11, 33], Groves-Manning [17], Osin [29], Yaman [38] and others. All these definitions except for those suggested in [29] are equivalent and require the group to be f...