Cannon-Thurston Maps for Pared Manifolds of Bounded Geometry

Abstract

Let N^h be a hyperbolic 3-manifold of bounded geometry corresponding to a hyperbolic structure on a pared manifold (M,P). Further, suppose that (\partial{M} - P) is incompressible, i.e. the boundary of M is incompressible away from cusps. Further, suppose that M_{gf} is a geometrically finite hyperbolic structure on (M,P). Then there is a Cannon- Thurston map from the limit set of M_{gf} to that of N^h. Further, the limit set of N^h is locally connected. This answers in part a question attributed to Thurston.

57 pages, 4 figures, Final version incorporating referee's comments. To appear in Geometry and Topology

References in corpus (6)

• Bounded geometry for Kleinian groups
• Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps
• The classification of Kleinian surface groups, II: The Ending Lamination Conjecture
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• Cannon-Thurston Maps and Bounded Geometry
• Blowing up and down compacta with geometrically finite convergence actions of a group
• Limits of limit sets II: Geometrically Infinite Groups
• On Cannon-Thurston maps for relatively hyperbolic groups
• Cannon-Thurston fibers for iwip automorphisms of $F_N$
• Conical limit points and the Cannon-Thurston map
• Tight trees and model geometries of surface bundles over graphs
• Measurable Rigidity for Kleinian groups
• Singular measures on the limit set of a Kleinian group
• Geometrically finite and infinite Kleinian Groups