Relatively hyperbolic metric bundles and Cannon-Thurston map

TL;DR

This paper investigates the relative hyperbolicity of metric bundles with hyperbolic fibers, establishing conditions for hyperbolicity and the existence of Cannon-Thurston maps, with applications to group extensions.

Contribution

It introduces conditions under which metric bundles with hyperbolic fibers are relatively hyperbolic and proves the existence of Cannon-Thurston maps in this setting.

Findings

Metric bundles are relatively hyperbolic under certain conditions.

Pullbacks of hyperbolic bundles via qi embeddings are relatively hyperbolic.

Existence of Cannon-Thurston maps for these bundles and group extensions.

Abstract

Given a metric (graph) bundle $X$ over $B$ where all the fibres are strongly relatively hyperbolic and nonelementary we show that, under certain conditions, $X$ is strongly hyperbolic relative to a collection of maximal cone-subbundles of horosphere-like spaces. Further, given a coarsely Lipschitz qi embedding $i: A\to B$, we show that the pullback $Y$ is strongly relatively hyperbolic and the map $Y\to X$ admits a Cannon-Thurston (CT) map. As an application, we prove a group-theoretic analogue of this result for a relatively hyperbolic extension of groups.