Homology directions and veering triangulations

The second real homology group of a hyperbolic 3-manifold M has a norm, called the Thurston norm, whose unit ball is a rational polyhedron detecting information about the topology of M. A collection of faces of this polyhedron called fibered faces organize all possible fibrations of M over the circle. The cone over a fibered face has a nice description, due to Fried, as the dual of the so-called cone of homology directions of a certain flow. In this relatively self-contained defense, we will show that this cone of homology directions is generated by a canonical family of curves living in one of Agol’s veering triangulations. This gives a new characterization of the cone over a fibered face. If time permits, we will discuss some related results.