Cannon-Thurston Maps for Kleinian Groups

TL;DR

This paper proves the existence of Cannon-Thurston maps for a broad class of Kleinian groups, confirming longstanding conjectures and elucidating the structure of pre-images related to ending laminations.

Contribution

It establishes the existence of Cannon-Thurston maps for all finitely generated Kleinian groups without parabolics, extending previous results and confirming conjectures of Thurston, McMullen, and Otal.

Findings

Cannon-Thurston maps exist for degenerate free groups without parabolics.

Pre-images of points under these maps correspond to ending lamination leaves.

Results confirm conjectures of Thurston, McMullen, and Otal.

Abstract

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 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 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 without parabolics.