We prove the existence of Cannon-Thurston maps for simply and doubly degenerate surface Kleinian groups. As a consequence we prove that connected limit sets of finitely generated Kleinian groups are locally connected.

We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the existence of quasi-isometric sections in this generality. Then we prove a combination theorem for metric (graph) bundles (including exact sequences of groups) that establishes sufficient conditions, particularly flaring, under which the metric bundles are hyperbolic. We use this to give examples of surface bundles over hyperbolic disks, whose universal cover is Gromov-hyperbolic. We also show that in typical situations, flaring is also a necessary condition.Comment: v3: Major revision: 56 pages 5 figures. Many details added. Characterization of convex cocompact subgroups of mapping class groups of surfaces with punctures in terms of relative hyperbolicity given v4: Final version incorporating referee comments: 63 pages 5 figures. To appear in Geom. Funct. Ana

We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn's Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Combination Theorem.20F67; 57M50

This is an expository paper. We prove the Cannon-Thurston property for bounded geometry surface groups with or without punctures. We prove three theorems, due to Cannon-Thurston, Minsky and Bowditch. The proofs are culled out of earlier work of the author. AMS subject classification = 20F32, 57M50Contents Let (X, d X ) be a hyperbolic metric space and Y be a subspace that is hyperbolic with the inherited path metric d Y . By adjoining the Gromov boundaries ∂X and ∂Y to X and Y , one obtains their compactifications X and Y respectively.Let i : Y → X denote inclusion.Definition: Let X and Y be hyperbolic metric spaces and i : Y → X be an embedding. A Cannon-Thurston mapî from Y to X is a continuous extension of i.

Abstract. In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups without accidental parabolics. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one.

The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map.

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Abstract. We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalizes a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.