Surjectivity of the Cannon--Thurston map in metric (graph) bundles

Abstract

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.