CANNON-THURSTON MAPS FOR TREES OF HYPERBOLIC METRIC SPACES

MAHAN MITRA

Abstract. Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasiisometrically 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 î : X̂_{v} → X̂. 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.