Cannon-Thurston Maps for Trees of Hyperbolic Metric Spaces

Original · EN

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 (Xᵥ,dᵥ) denote the hyperbolic metric space corresponding to v. Then i: Xᵥ → X extends continuously to a map i: Xᵥ → 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.