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

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.