Pseudo-Anosov Reeb flows and contact structures

Abstract

We introduce the notion of a pseudo-Anosov contact structure, which admits a type of singular contact form with pseudo-Anosov Reeb flow. We prove that contact homology detects the free homotopy classes of closed orbits of any pseudo-Anosov Reeb flow and that any pseudo-Anosov contact structure is universally tight and torsion free. Many applications are given, including new cases of the Finiteness Conjecture for transitive pseudo-Anosov flows. Our proofs use a flavor of contact homology graded by a free homotopy class of loops, defined for any contact manifold. We establish several properties of this type of contact homology that may be of independent interest.