The Cannon-Thurston map is surjective for metric graph bundles with hyperbolic fibers that are either one-ended or of bounded valence with coarsely surjective barycenter maps.
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Surjectivity of the Cannon--Thurston map in metric (graph) bundles
The Cannon-Thurston map is surjective for metric graph bundles with hyperbolic fibers that are either one-ended or of bounded valence with coarsely surjective barycenter maps.