Ending Laminations and Cannon-Thurston Maps

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. In conjunction with the Ending Lamination Conjecture of Thurston (now Theorem of Minsky, Masur-Minsky, Brock-Canary-Minsky), the main result of this paper gives a complete description of the isometry type of a simply degenerate hyperbolic manifold homeomorphic to S × [0,∞) in terms of the action of its fundamental group on the limit set and the domain of discontinuity. AMS Subject Classification: 57M50