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Semiconjugacies between Kleinian group actions on the Riemann sphere. (English) Zbl 1011.30035
The author discusses the action of a geometrically infinite Kleinian group $$\Gamma$$ on the Riemann sphere and shows that in some conditions the semiconjugacy with the action of a geometrically finite Kleinian group is determined by the end invariants of $$\Gamma$$. With respect to a semiconjugacy this discussion is related to the extension of a map of hyperbolic 3-space continuously to the boundary at infinity, that is the Riemann sphere. More generally, the author discusses the extension problem in the Gromov-hyperbolic spaces and gives a sufficient condition for a map between Gromov-hyperbolic spaces to be extend continuously to their boundaries.

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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