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