A Combination Theorem for Metric Bundles

TL;DR

This paper introduces a new framework for metric bundles, proving conditions like flaring that ensure hyperbolicity, and provides examples such as hyperbolic surface bundles, advancing the understanding of coarse geometric structures.

Contribution

It generalizes trees of metric spaces to metric bundles, proves the existence of quasi-isometric sections, and establishes a combination theorem for hyperbolicity under flaring conditions.

Findings

Existence of quasi-isometric sections in metric bundles

Hyperbolicity of bundles under flaring conditions

Examples of hyperbolic surface bundles over disks

Abstract

We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the existence of quasi-isometric sections in this generality. Then we prove a combination theorem for metric (graph) bundles (including exact sequences of groups) that establishes sufficient conditions, particularly flaring, under which the metric bundles are hyperbolic. We use this to give examples of surface bundles over hyperbolic disks, whose universal cover is Gromov-hyperbolic. We also show that in typical situations, flaring is also a necessary condition.