From veering triangulations to Cannon-Thurston maps

Abstract: Agol introduced veering triangulations of mapping tori, whose combinatorics are canonically associated to the pseudo-Anosov monodromy. In previous work, Hodgson, Rubinstein, Tillmann and I found examples of veering triangulations that are not layered and therefore do not come from Agol's construction.

However, non-layered veering triangulations retain many of the good properties enjoyed by mapping tori. For example, Schleimer and I constructed a canonical circular ordering of the cusps of the universal cover of a veering triangulation. Its order completion gives the veering circle; collapsing a pair of canonically defined laminations gives a surjection onto the veering sphere.

In work in progress, Manning, Schleimer, and I prove that the veering sphere is the Bowditch boundary of the manifold's fundamental group (with respect to its cusp groups). As an application we produce Cannon-Thurston maps for all veering triangulations. This gives the first examples of Cannon-Thurston maps that do not come, even virtually, from surface subgroups.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( slides )

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