Cannon-Thurston maps for pared manifolds of bounded geometry

Abstract

Let N^h ∈, H(M, P) be a hyperbolic structure of bounded geometry on a pared manifold such that each component of ∂₀M = ∂M - P is incompressible. We show that the limit set of N^h is locally connected by constructing a natural Cannon-Thurston map. This provides a unified treatment, an alternate proof and a generalization of results due to Cannon and Thurston, Minsky, Bowditch, Klarreich and the author.