Geometry of foliations and flows I: Almost transverse pseudo-Anosov flows and asymptotic behavior of foliations

Abstract

Let $\mathcal{F}$ be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that $\mathcalF$ is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity; therefore the limit sets of the leaves are continuous images of the circle. One important corollary is that if $\mathcal{F}$ is a Reebless, infinite depth foliation in a hyperbolic 3-manifold, then it has the continuous extension property. Such infinite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classes of foliations, including a large class of foliations where all leaves are dense, and infinitely many examples with one sided branching. One extremely useful tool is a detailed understanding of the topological structure and asymptotic properties of the 1-dimensional foliations in the leaves of e $\mathcal{F}$ induced by the stable and unstable foliations of the flow.