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

Abstract

Let F1 and F2 be transverse two dimensional foliations with Gromov hyperbolic leaves in a closed 3-manifold M whose fundamental group is not solvable, and let G be the one dimensional foliation obtained by intersection. We show that G is \emph{leafwise quasigeodesic} in F1 and F2 if and only if the foliation GL induced by G in the universal cover L of any leaf of F1 or F2 has Hausdorff leaf space. We end up with a discussion on the hypothesis of Gromov hyperbolicity of the leaves.