Semiconjugacies Between Relatively Hyperbolic Boundaries

TL;DR

This paper establishes the existence and structure of Cannon-Thurston maps for certain Kleinian groups, linking boundary behavior to ending laminations in the context of relatively hyperbolic groups.

Contribution

It proves the existence of Cannon-Thurston maps for pared manifolds with incompressible boundaries and describes their structure via ending laminations.

Findings

Existence of Cannon-Thurston maps for specific Kleinian groups.

Structural description of these maps in terms of ending laminations.

Extension of boundary map theory to relatively hyperbolic groups.

Abstract

We prove the existence of Cannon-Thurston maps for Kleinian groups corresponding to pared manifolds whose boundary is incompressible away from cusps. We also describe the structure of these maps in terms of ending laminations.