We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary.Using the Floyd completion we further prove that the property of relative hyperbolicity is invariant under quasi-isometric maps. If a finitely generated group H admits a quasiisometric map ϕ into a relatively hyperbolic group G then H is itself relatively hyperbolic with respect to a system of subgroups whose image under ϕ is situated within a uniformly bounded distance from the parabolic subgroups of G.We then generalize the latter result to the case when ϕ is an α-isometric map for any polynomial distortion function α.As an application of our method we provide in the Appendix a new short proof of a basic theorem of Bowditch characterizing hyperbolicity.

We show that the right-angled hyperbolic polyhedra of finite volume in the hyperbolic space H n may only exist if n ≤ 14. We also provide a family of such polyhedra of dimensions n = 3, 4,. .. , 8. We prove that for n = 3, 4 the members of this family have the minimal total number of hyperfaces and cusps among all hyperbolic right-angled polyhedra of the corresponding dimension. This fact is used in the proof of the main result.

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For finitely supported random walks on finitely generated groups G we prove that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This yields new results for relatively hyperbolic groups. Our key estimate relates the Green and Floyd metrics, generalizing results of Ancona for random walks on hyperbolic groups and of Karlsson for quasigeodesics. We then apply these techniques to obtain some results concerning the harmonic measure on the limit sets of geometrically finite isometry groups of Gromov hyperbolic spaces. .

ABSTRACT. We study different notions of quasiconvexity for a subgroup H of a relatively hyperbolic group G. The first result establishes equivalent conditions for H to be relatively quasiconvex. As a corollary we obtain that the relative quasiconvexity is equivalent to the dynamical quasiconvexity. This answers to a question posed by D. Osin [Os06].In the second part of the paper we prove that a subgroup H of a finitely generated relatively hyperbolic group G acts cocompactly outside its limit set if and only if it is (absolutely) quasiconvex and every its infinite intersection with a parabolic subgroup of G has finite index in the parabolic subgroup.We then obtain a list of different subgroup properties and establish relations between them.