R-covered foliations and transverse pseudo-Anosov flows in atoroidal pieces

TL;DR

This paper investigates the geometric and dynamical properties of R-covered foliations with hyperbolic leaves in 3-manifolds, establishing the existence of transverse pseudo-Anosov flows in atoroidal pieces and analyzing their universal circle actions.

Contribution

It proves the existence of regulating pseudo-Anosov flows transverse to R-covered foliations with hyperbolic leaves in atoroidal 3-manifolds, and describes their universal circle actions.

Findings

Existence of regulating pseudo-Anosov flows in atoroidal pieces.

Description of deck transformations on the universal circle.

Construction of a regulating flow for the entire foliation.

Abstract

We study the transverse geometric behavior of 2-dimensional foliations in 3-manifolds. We show that an R-covered transversely orientable foliation with Gromov hyperbolic leaves in a closed 3-manifold admits a regulating, transverse pseudo-Anosov flow (in the appropriate sense) in each atoroidal piece of the manifold. The flow is a blow of a one prong pseudo-Anosov flow. In addition we show that there is a regulating flow for the whole foliation. We also determine how deck transformations act on the universal circle of the foliation.