Pseudo-Anosov flows on hyperbolic L-spaces

TL;DR

This paper constructs hyperbolic L-spaces with any number of distinct, non-orbit equivalent pseudo-Anosov flows, revealing complex interactions between flows, contact structures, and topological properties.

Contribution

It demonstrates the existence of hyperbolic L-spaces with multiple non-equivalent pseudo-Anosov flows and explores their implications for contact structures and flow dynamics.

Findings

Existence of hyperbolic L-spaces with n pseudo-Anosov flows for any n

Flows are quasigeodesic with no perfect fits

Flows induce n universally tight contact structures

Abstract

We prove that for each $n\in\mathbb{N}$ there is a hyperbolic L-space with $n$ pseudo-Anosov flows, no two of which are orbit equivalent. These flows have no perfect fits and are thus quasigeodesic. In addition, our flows admit positive Birkhoff sections, which we argue implies that they give rise to $n$ universally tight contact structures whose lifts to any finite cover are non-contactomorphic. This argument involves cylindrical contact homology together with the work of Barthelm\'e, Frankel, and Mann on the reconstruction of pseudo-Anosov flows from their closed orbits. These results answer more general versions of questions posed by Calegari and by Min and Nonino.