Cannon–Thurston maps for pared manifolds of bounded geometry

Abstract

Let $N^h\in H\left(M,P\right)$ be a hyperbolic structure of bounded geometry on a pared manifold such that each component of ${\partial }_{0}M=\partial M-P$ 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.