$\mathbb{R}$-covered foliations and transverse pseudo-Anosov flows in atoroidal pieces

Abstract

We study the transverse geometric behavior of 2-dimensional foliations in 3-manifolds. We show that an $\mathbb{R}$-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 up 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.