2025/07/09 by Rakesh Halder, Halder, Rakesh
#math.GT #math.MG
paper · pdf · doi:10.48550/arxiv.2507.07076
Metric (graph) bundles generalize the notion of fiber bundles to the context of geometric group theory and were introduced by Mj and Sardar. Suppose X is a metric (graph) bundle over B such that the fibers are (uniformly) hyperbolic, and the total space X is also hyperbolic. In this generality, Mj--Sardar proved that the inclusion of a fiber into X admits a continuous extension to the (Gromov) boundary. In this article, we prove that such a continuous extension map between boundaries is surjective in the following two key settings. (1) The fibers are uniformly quasiisometric to a nonelementary hyperbolic group. (2) The fibers are one-ended hyperbolic metric spaces. Our result generalizes a theorem of Bowditch in which the fibers were assumed to be the hyperbolic plane, and it answers a question posed by Lazarovich, Margolis and Mj.