Foliations, flows, and 3-manifolds: Topology and geometry

Washington University, Saint Louis MO

Investigators

Abstract

Proposal: DMS-0071683 Project title: "Foliations, flows and 3-manifolds: Topology and geometry" PI: Sergio R. Fenley ABSTRACT: Reebless foliations are a basic and fundamental object in the study of 3-manifolds. They yield deep results on 3-manifold topology and are also related to the geometrization conjecture for 3-manifolds. An important tool is the universal circle associated to a foliation with hyperbolic leaves (the generic case). Using this tool, the PI has recently proved that in the case of R-covered foliations, then either the manifold is toroidal or there is a pseudo-Anosov flow transverse to the foliation. Such a flow has excellent dynamical properties and this gives a strong relationship with the topology of the manifold. One objective of this project is to analyze the universal circle construction for various families of foliations and to search for transverse pseudo-Anosov flows. A second goal is to study foliation and transverse flows as dual objects and to understand the deep consequences of their joint dynamical structure. The project aims to study general pseudo-Anosov flows and the additional structure imposed by a transverse foliation. A third goal of the project is to understand geometric behavior of foliations and transverse flows in hyperbolic 3-manifolds - the generic case. The focus will be on the large scale geometric behavior in the universal cover. An important question to be analyzed is whether the pseudo-Anosov flows transverse to foliations are quasigeodesic - that is, whether they measure distances well. This property has been proved by the PI (and Lee Mosher) in the case of a flow transverse to a Reebless finite depth foliation. One objective of this study is to use the quasigeodesic property for flows to derive information about the asymptotic geometric behavior of the foliation transverse to the flow. This is specially promising in the case of general finite depth foliations. A 3-manifold is an object that locally has 3-dimensions, like 3-dimensional Euclidean space. A 2-dimensional foliation of a3-manifold is a decomposition of the manifold into 2-dimensional objects, much like the pages of a book. Good foliations exist in large classes of 3-manifolds and they yield very useful information about the manifold. The goal of the project is to understand the relationship of the foliation with geometric and topological structures of the manifold. Geometry measures distance and roughly topology measures the structures of "holes" in the manifold. It is best to look at the universal cover of the manifold: for example the universal cover of a cylinder is obtained by unrolling it into an infinite carpet. Universal covers are unbounded objects but they carry a lot of information about the manifold, contained in the "folding" allowed to recover the manifold. One important goal of the project is to analyse the foliation in the universal cover. Of particular interest is the large scale properties of the "leaves" of the foliation. We will concentrate in the case where the manifold is hyperbolic -this is the generic situation. Finally there is a dual object to a foliation which is a transverse flow to the leaves of the foliation. The flow and foliation jointly produce a rich dynamical structure in the manifold with many deep consequences. There are many cases when there is a "tightest" amongst all possible transverse flows. The tightest flow should give information about the structure of the foliation and connections with the manifold. The tightest flow is called a pseudo-Anosov flow.