Semiconjugacies between Kleinian group actions on the Riemann sphere

We consider the problem of characterizing topologically the action of a Kleinian group on the Riemann sphere Ĉ. We prove that certain geometrically infinite Kleinian group actions on Ĉ can be obtained from geometrically finite ones by a semiconjugacy that is determined by the end invariants of the geometrically infinite group. This turns out to be related to the problem of continuous extensions of maps of hyperbolic 3-space H³ to maps of its boundary at infinity, Ĉ. Along the way we consider the general problem of extending maps to the boundary at infinity in Gromov-hyperbolic metric spaces. We give criteria for extending to the boundary a map h: X → Y between hyperbolic spaces if it extends to the boundary of certain subsets of X.