Intersection of transverse foliations in 3-manifolds: Hausdorff leafspace implies leafwise quasi-geodesic

TL;DR

This paper establishes a link between the Hausdorff property of leaf spaces in universal covers and the leafwise quasigeodesic nature of intersection foliations in certain 3-manifolds with hyperbolic leaves.

Contribution

It proves that the intersection foliation is leafwise quasigeodesic if and only if the induced foliation in universal covers has a Hausdorff leaf space, under specific hyperbolicity conditions.

Findings

Leafwise quasigeodesic property characterized by Hausdorff leaf spaces.

Equivalence established for transverse foliations with hyperbolic leaves.

Discussion on the necessity of Gromov hyperbolicity hypothesis.

Abstract

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be transverse two dimensional foliations with Gromov hyperbolic leaves in a closed 3-manifold $M$ whose fundamental group is not solvable, and let $\mathcal{G}$ be the one dimensional foliation obtained by intersection. We show that $\mathcal{G}$ is \emph{leafwise quasigeodesic} in $\mathcal{F}_1$ and $\mathcal{F}_2$ if and only if the foliation $\mathcal{G}_L$ induced by $\mathcal{G}$ in the universal cover $L$ of any leaf of $\mathcal{F}_1$ or $\mathcal{F}_2$ has Hausdorff leaf space. We end up with a discussion on the hypothesis of Gromov hyperbolicity of the leaves.