Cannon-Thurston Maps for Kleinian Groups

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.

39 pgs 1 fig. Final version incorporating referee comments. To appear in Forum of Mathematics, Pi

References in corpus (5)

• Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps
• Geometric inflexibility and 3-manifolds that fiber over the circle
• On Discreteness of Commensurators
• Ending Laminations and Cannon-Thurston Maps
• Commensurators of non-free finitely generated Kleinian groups

Cited by in corpus (4)

• Algebraic Ending Laminations and Quasiconvexity
• Combining Rational maps and Kleinian groups via orbit equivalence
• Discontinuous motions of limit sets
• Combination Theorems in Groups, Geometry and Dynamics