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

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> \mathcal{F}_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> \mathcal{F}_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \mathcal{F}_{2} if and only if the foliation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">G</m:mi> <m:mi>L</m:mi> </m:msub> </m:math> \mathcal{G}_{L} induced by 𝒢 in the universal cover 𝐿 of any leaf of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> \mathcal{F}_{1} or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \mathcal{F}_{2} has Hausdorff leaf space. We end up with a discussion on the hypothesis of Gromov hyperbolicity of the leaves.