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

TL;DR

This paper proves the surjectivity of the Cannon--Thurston boundary extension in metric bundle spaces with hyperbolic fibers, generalizing previous results and answering open questions in geometric group theory.

Contribution

It establishes surjectivity of boundary maps for hyperbolic fibers in metric bundles, extending Bowditch's theorem to broader fiber types.

Findings

Surjectivity of boundary extension in specific hyperbolic fiber settings.

Generalization of Bowditch's theorem to new fiber classes.

Answers to open questions in the theory of 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.