Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let v be a vertex of T . Let (X v , d v ) denote the hyperbolic metric space corresponding to v. Then i : X v → X extends continuously to a mapî : X v → X. This generalizes a Theorem of Cannon and Thurston. The techniques are used to give a new proof of a result of Minsky: Thurston's ending lamination conjecture for certain Kleinian groups. Applications to graphs of hyperbolic groups and local connectivity of limit sets of Kleinian groups are also given.

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Abstract. We say that the width of an infinite subgroup H in G is n if there exists a collection of n essentially distinct conjugates of H such that the intersection of any two elements of the collection is infinite and n is maximal possible. We define the width of a finite subgroup to be 0. We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically finite surfaces in closed hyperbolic 3-manifolds satisfy the k-plane property for some k.

Abstract. Let 1 → H → G → Z → 1 be an exact sequence of hyperbolic groups induced by an automorphism φ of the free group H. Let H 1 (⊂ H) be a finitely generated distorted subgroup of G. Then there exist N > 0 and a free factor K of H such that the conjugacy class of K is preserved by φ N and H 1 contains a finite index subgroup of a conjugate of K. This is an analog of a theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.