Ending Laminations and Cannon-Thurston Maps

Abstract

In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one. This completes the proof of the conjectural picture of Cannon-Thurston maps for surface groups.

v4: Final version 22pgs 2figures. Includes the main theorem of the appendix arXiv:1002.2090 by Shubhabrata Das and Mahan Mj. To appear in Geometric and Functional Analysis

References in corpus (2)

• Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps
• On Discreteness of Commensurators

Cited by in corpus (6)

• Algebraic Ending Laminations and Quasiconvexity
• Cannon-Thurston Maps for Kleinian Groups
• Limits of limit sets II: Geometrically Infinite Groups
• Combining Rational maps and Kleinian groups via orbit equivalence
• Discontinuous motions of limit sets
• Combination Theorems in Groups, Geometry and Dynamics