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 $i:X_v\to X$ extends continuously to a map $\hat{i}:\widehat{X_v}\to \hat{X}$. 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.