Cannon-Thurston Maps for Trees of Hyperbolic Metric Spaces

Details Cannon-Thurston Maps for Trees of Hyperbolic Metric Spaces Cannon-Thurston Maps for Trees of Hyperbolic Metric Spaces

1996-09-23

arxiv.org/abs/math/9609209v1

J. Differential Geom. 48 (1998), no. 1, 135-164

Mathematics

Geometric Topology

Scientific paper

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.

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