Mathematical Sciences: Essential Laminations in 3-Manifolds

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Essential laminations are a recently defined generalization of both the incompressible surface and the codimension-one foliation without Reeb components of a compact 3-manifold. Both of these more classical objects have proved remarkably useful in obtaining geometric/topological information about 3-manifolds from homotopy-theoretic data, and the essential lamination has already begun to show itself to be a worthy successor to both of its `parents.' The investigator intends to use essential laminations to study the extent to which the fundamental group of a 3-manifold determines the manifold up to homeomorphism, and also to study the question of whether or not the manifold admits a hyperbolic structure. He further intends to use the existence of normal forms for essential laminations to study the question of whether or not a 3-manifold (principally a hyperbolic 3-manifold) contains an essential lamination. It is a surprising fact that although we live in a three dimensional space, a so-called 3-manifold, and so are blessed with a natural intuition about such geometric objects, in the end this does not carry us as far as we might have expected, for questions which have been settled by algebraic calculations for higher dimensional manifolds still remain baffling in the 3-dimensional case. The most famous of these is the celebrated conjecture of Poincare from around the turn of the century concerning 3- dimensional spheres, where precisely the original 3-dimensional case is the only one still open. The investigator is pursuing a variety of questions about 3-dimensional manifolds, some with slightly strange notions of distance on them, so-called hyperbolic metrics, but time and time again these questions have been shown to have clear relevance to the case of manifolds with a more familiar notion of distance.