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

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.