From veering triangulations to convergence actions and back again

Jason Fox Manning; Saul Schleimer; Henry Segerman

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arXiv · 2609.14598

From veering triangulations to convergence actions and back again

Jason Fox Manning·Saul Schleimer·Henry Segerman

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Abstract

Suppose that $M$ is a finite-volume cusped hyperbolic three-manifold, equipped with a veering triangulation $\mathcal{V}$. We prove that the action of the fundamental group of $M$ on the veering two-sphere is a geometrically finite convergence action. Applying a result of Yaman, we deduce that the veering two-sphere is equivariantly homeomorphic to the boundary of hyperbolic space. As an application, we obtain Cannon-Thurston maps associated to veering triangulations. If $\mathcal{V}$ is layered we recover the classical Cannon-Thurston map. If it is not we obtain Cannon-Thurston maps that do not come from surface subgroups. These are the first such examples in the cusped case. Finally, we implement an algorithm to draw approximations of these Cannon-Thurston maps. This improves upon previous approximations obtained by Thurston and others.

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BibTeX RIS

Jason Fox Manning, Saul Schleimer, Henry Segerman. 2026-09-13. From veering triangulations to convergence actions and back again. https://arxiv.org/abs/2609.14598

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