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Cannon-Thurston maps for Kleinian groups. (English) Zbl 1370.57008
Question 14 of Thurston’s famous and most influential problem list on hyperbolic 3-manifolds and Kleinian groups can be rephrased as follows. Suppose that $$\Gamma$$ is a geometrically finite Kleinian group and $$G$$ an arbitrary Kleinian group abstractly isomorphic to $$\Gamma$$, by an isomorphism preserving parabolics. Then does there exist a continuous map from the set of limit points of $$\Gamma$$ to the set of limit points of $$G$$ taking a fixed point of an element of $$\Gamma$$ to a fixed point of the corresponing element of $$G$$? Such a continuous map is called a Cannon-Thurston map, cf. the paper of J. W. Cannon and W. P. Thurston [Geom. Topol. 11, 1315–1355 (2007; Zbl 1136.57009)], based on a preprint from 1985.
In a previous paper [Ann. Math. (2) 179, No. 1, 1–80 (2014; Zbl 1301.57013)], the present author showed that Cannon-Thurston maps exist for simply or doubly degenerate surface Kleinian groups without accidental parabolics, and in a second paper [Geom. Funct. Anal. 24, No. 1, 297–321 (2014; Zbl 1297.57040)] the author showed that point pre-images of the Cannon-Thurston map for such groups correspond to endpoints of leaves of ending laminations whenever a point has more than one pre-image.
“The aim of this paper is to apply the techniques developed in these two papers to extend these results to arbitrary finitely generated Kleinian groups without parabolics”. “We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon-Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon-Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.”

MSC:
 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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References:
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