A complex hyperbolic quasi-Fuchsian group is a discrete, faithful, type preserving and geometrically finite representation of a surface group as a subgroup of the group of holomorphic isometries of complex hyperbolic space. Such groups are direct complex hyperbolic generalisations of quasi-Fuchsian groups in three dimensional (real) hyperbolic geometry. In this article we present the current st...

We show that for any non–elementary hyperbolic group H and any finitely presented group Q, there exists a short exact sequence 1 → N → G → Q → 1, where G is a hyperbolic group and N is a quotient group of H . As an application we construct a representation rigid but not superrigid hyperbolic group, show that adding relations of the form xn = 1 to a presentation of a hyperbolic group may drastic...

We show that for any non–elementary hyperbolic group H and any finitely presented group Q, there exists a short exact sequence 1 → N → G → Q → 1, where G is a hyperbolic group and N is a quotient group of H . As an application we construct a hyperbolic group that has the same n–dimensional complex representations as a given finitely generated group, show that adding relations of the form xn = 1...

We show that for any non–elementary hyperbolic group H and any finitely presented group Q, there exists a short exact sequence 1 → N → G → Q → 1, where G is a hyperbolic group and N is a quotient group of H . As an application we construct a representation rigid but not superrigid hyperbolic group, show that adding relations of the form xn = 1 to a presentation of a hyperbolic group may drastic...

This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a 2-dimensional Artin group the Deligne complex is Gromov hyperbolic preci...

1. If a closed irreducible 3-manifold has word hyperbolic fundamental group, does it admit a Riemannian metric of constant negative curvature? 2. Is every nitely presented subgroup of a word hyperbolic group word hyperbolic? (Noel Brady Brady] has spoken on an example of nitely presented subgroup H of a word hyperbolic group G where cd(G) = 3 and H is not hyperbolic. I had shown previously that...

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. • If H is a non-parabolic subgroup of a relatively hyperb...

In [12], I. Kapovich and P. Schupp showed that certain 2-dimensional Artin groups (those with all relator indices at least 7) are hyperbolic relative (in the sense of Farb) to their non-free rank 2 parabolic subgroups. This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of the associated Deligne complex. We prove a relative version of the Mi...

We discuss the notion of the universal relatively hyperbolic structure on a group which is used in order to characterize relatively hyperbolic structures on the group. We also study relations between relatively hyperbolic structures on a group and relative quasiconvexity for subgroups of the group.

We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling.