Foliations and Geometry

Bill Thurston
(MSRI & UC Davis)

Saturday, February 8, 1997
2:30 PM

I will discuss progress in understanding the geometry of foliations, particularly in dimension 3, in connection with the program of trying to reconcile and combine several approaches toward three-dimensional topology. A main goal of the program is to find a construction for a geometric decomposition for a 3-manifold based on a taut foliation, an essential lamination, or a tight contact structure. An important related goal is to analyze how these structures relate to a hyperbolic structure. More specifically, I will touch on the following points, discussing some of this in depth. Some of this is quite new, and subject to change before the talk.
  1. The pocket principal. By looking at diffusion along the leaves of a foliation (and/or symmetric random walks on groups or pseudo-groups of homeomorphisms), one finds that nearby leaves of a codimension-one foliation mostly converge at infinity, in a sense that can be made precise. A good image is that nearby leaves in the universal cover form pockets, with openings that are small compared to the seams that join them.
  2. The geodesic flow of leaves. It is known that foliations with 2-dimensional leaves, under reasonable general circumstances, have Riemannian metrics which are negatively-curved on each leaf. The pocket principal gives a way to compare geodesic flows on different leaves.
  3. A universal circle at infinity. When a 3-manifold has a codimension-one foliation with negatively-curved leaves, a canonical universal circle at infinity can be constructed, such that the circle at infinity for any particular leaf is identified either with the universal circle, or with a quotient of it.
  4. Transverse laminations. The pseudo-Anosov theory for diffeomorphisms of surfaces has previously been extended to give pseudo-Anosov flows transverse to certain foliations. This theory extends more generally, to usually give a pair of essential laminations transverse to a taut foliation.
  5. Iteration on conformal structures. There is an iterative process associated with a foliation or lamination, generalizing the "skinning map" construction for Haken 3-manifolds and related to a similar process for iterated rational maps, which I conjecture will converge in the case of a sincere essential lamination to give a geometric decomposition for the underlying manifold (and which conjecturally yields other information in the case of foliations). (A lamination is only sincere if it can't be derived from some foliation merely by removing I-bundle neighborhoods of leaves).
  6. Groups acting on S1, Peano curves, and ending laminations.
  7. Further prospects: tight contact structures, essential branched surfaces, word-hyperbolic groups, automatic groups, and conformal structures at infinity.
    Back to Winter 1997 meeting of the PNGS.

    Suggestions or corrections to

    Jack Lee <lee@math.washington.edu>.