Veering triangulations and Cannon-Thurston maps

TL;DR

This paper explores how veering triangulations influence Cannon-Thurston maps in hyperbolic surface bundles, establishing a connection between the triangulation and the filling order of the sphere by the surjection.

Contribution

It demonstrates that the natural triangulation of the surface bundle determines the order of the Cannon-Thurston map filling the sphere, under specific singularity conditions.

Findings

The filling order of the sphere is dictated by the veering triangulation.

The triangulation introduced by Agol is key to understanding the Cannon-Thurston map.

The results apply when all singularities are at punctures of the fiber.

Abstract

Any hyperbolic surface bundle over the circle gives rise to a continuous surjection from the circle to the sphere, by work of Cannon and Thurston. We prove that the order in which this surjection fills out the sphere is dictated by a natural triangulation of the surface bundle (introduced by Agol) when all singularities of the invariant foliations are at punctures of the fiber.