Intersection of transverse foliations in 3-manifolds: Hausdorff leaf space implies leafwise quasigeodesic

Abstract

Let

and

F 2 \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 𝒢 be the one-dimensional foliation obtained by intersection. We show that 𝒢 is leafwise quasigeodesic in

F 1 \mathcal{F}_{1}

and

F 2 \mathcal{F}_{2}

if and only if the foliation

G L \mathcal{G}_{L}

induced by 𝒢 in the universal cover 𝐿 of any leaf of

F 1 \mathcal{F}_{1}

or

F 2 \mathcal{F}_{2}

has Hausdorff leaf space. We end up with a discussion on the hypothesis of Gromov hyperbolicity of the leaves.