  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.
