Pseudo-Anosov Reeb flows and contact structures

TL;DR

This paper introduces pseudo-Anosov contact structures with singular contact forms and pseudo-Anosov Reeb flows, demonstrating their tightness, torsion-freeness, and the ability of contact homology to detect orbit classes, with applications to the Finiteness Conjecture.

Contribution

It defines pseudo-Anosov contact structures and develops contact homology graded by free homotopy classes, providing new tools and results in contact topology.

Findings

Contact homology detects free homotopy classes of closed orbits.

Pseudo-Anosov contact structures are universally tight and torsion free.

Applications include new cases of the Finiteness Conjecture for pseudo-Anosov flows.

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.