A Combination Theorem for Metric Bundles

Details A Combination Theorem for Metric Bundles A Combination Theorem for Metric Bundles

2009-12-14

arxiv.org/abs/0912.2715v3

Mathematics

Geometric Topology

v3: Major revision: 56 pages 5 figures. Many details added. Characterization of convex cocompact subgroups of mapping class gr

Scientific paper

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.

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