Geometry of foliations and flows
by
Sergio Fenley
Florida State University

Let F be a Reebless, finite depth foliation in a closed 3-manifold with negatively curved fundamental group . Such foliations exist whenever the second homology is non trivial. We show that the leaves in the universal cover extend continuously to the sphere at infinity, hence the limit sets are continuous images of the circle. This follows from a more general general result, which proves the continuous extension property whenever a foliation in such 3-manifolds is almost transverse to a quasigeodesic pseudo-Anosov flow. This applies to other classes of foliations, including a large class of foliations where all leaves are dense. One key technical tool is a detailed understanding of asymptotic properties of almost pseudo-Anosov singular 1-dimensional foliations in the leaves of F lifted to the universal cover.

Date received: March 13, 2008