Faces of the Thurston norm ball up to isotopy

Abstract:

Let \(M\) be a three-manifold with nondegenerate Thurston norm \(x\) on its second homology. There is a partial dictionary between the combinatorics of the polyhedral unit ball of \(x\) and the topological features of \(M\). This dictionary is quite incomplete, but its existing entries are tantalizing.

Currently, most of the entries of this dictionary concern fibered faces of the unit ball. Thurston proved that these organize all fibrations of \(M\) over the circle. Fried and Mosher tell us more: for each fibered face \(F\) there is a (canonical) pseudo-Anosov flow whose Euler class computes the norm \(x\) in the cone over \(F\). Furthermore, the flow "sees" certain least-complexity surfaces. Further work of Mosher shows that, under certain conditions, pseudo-Anosov flows can naturally specify nonfibered faces of the unit ball.

After giving some of this background I will discuss results from my recent preprint (see link). I show that Agol's veering triangulations can be used to determine faces of Thurston norm balls, to compute the Thurston norm over those faces, and to collate all isotopy classes of least-complexity surfaces over those faces. This analysis includes nonfibered faces.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the discipline

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