Finiteness of Anosov Flows on Closed Hyperbolic 3-manifolds via Contact Geometry

We show that on every closed 3-manifold, the pair of oppositely oriented contact structures associated with an Anosov flow completely determines the free homotopy data of its periodic orbits.

As a consequence, using works of Barthlemé-Frankel-Mann and C.-Giroux-Honda, we obtain that the number isotopy orbit equivalence classes of Anosov flows on a closed hyperbolic 3-manifold is finite.

We will also discuss further results involving toroidal manifolds as well as pseudo-Anosov flows.

This is joint work with Jonathan Bowden.