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CANNON–THURSTON MAPS FOR KLEINIAN GROUPS

Part of: Lie groups Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  22 May 2017

MAHAN MJ
MAHAN MJ*
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India; [email protected], [email protected]

Abstract

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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 classification

Primary: 57M50: Geometric structures on low-dimensional manifolds 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F65: Geometric group theory 22E40: Discrete subgroups of Lie groups
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 5 , 2017 , e1
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2017

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