Cannon-Thurston Maps for Kleinian Groups

Details Cannon-Thurston Maps for Kleinian Groups Cannon-Thurston Maps for Kleinian Groups

2010-02-04

arxiv.org/abs/1002.0996v3

Mathematics

Geometric Topology

28 pages 1 figure. The first part of this paper was contained in an earlier version of arXiv:math/0607509. The second part of

Scientific paper

We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. 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, proving a conjecture of McMullen. We also show that point pre-images under Cannon-Thurston maps for degenerate free groups without parabolics correspond to end-points 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.

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