Cyril Lecuire

We provide a proof that the classes of finitely generated Kleinian groups and of three-manifold groups are quasi-isometrically rigid.

We study horospheres in hyperbolic 3-manifolds M all whose ends are degenerate. Deciding which horospheres in M are properly embedded and which are dense reduces to a) studying the horospherical limit set; b) deciding which almost minimizing geodesics in M go through arbitrarily thin parts. As an answer to (a), we show that the horospherical limit set consists precisely of the injective points of the Cannon-Thurston map. Addressing (b), we provide characterizations, sufficient conditions as well as a number of examples and counterexamples.