Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps

TL;DR

This paper establishes the existence of continuous boundary maps for inclusions in trees of relatively hyperbolic spaces, generalizing previous results for hyperbolic groups and 3-manifolds.

Contribution

It extends Cannon-Thurston map existence to a broad class of relatively hyperbolic spaces and groups under qi-embedded conditions.

Findings

Proves continuous boundary extensions for vertex space inclusions.

Generalizes Bowditch's and Mitra's results to relatively hyperbolic settings.

Applies to finite graphs of relatively hyperbolic groups.

Abstract

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.