Cannon-Thurston maps for trees of hyperbolic metric spaces

Abstract

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 map i ^ : 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.