Essential Laminations in 3-Manifolds

It is a well-established principle in manifold topology that by studying codimension-1 objects and their complementary pieces one obtains a great deal of topological and geometric information about the manifold itself. This is particularly true for 3-dimensional manifolds. Using incompressible surfaces a great many advances have been made during the last 30 years; the most spectacular results were obtained by Haken [Ha], Waldhausen [W] and Thurston [T1]. More recently, work [T2], [G14] using taut foliations in 3-manifolds has proven fruitful. Unfortunately (in some sense) most closed 3-manifolds do not contain incompressible surfaces and it is currently not known exactly which 3-manifolds have taut foliations. The purpose of this paper is to study another codimension-1 object, the lamination, which is a generalization of the incompressible surface as well as the taut foliation. We will show that the existence of an lamination in a 3-manifold M implies that M has useful properties similar to those of manifolds having either an incompressible surface or a taut foliation. We will also show that the universal cover of a manifold containing an lamination is R3. Precise definitions of terms used in the following theorem will be given in Section 1. Here we give only a rough idea. We begin with two alternative approximate definitions of essential lamination. (I) An lamination in a 3-manifold M is a lamination satisfying four conditions: The inclusion of leaves of the lamination into M should induce an injection on sTJ, the complement of the lamination should be irreducible, no leaf should be a sphere, and the lamination should be end-incompressible. To say that a lamination is end-incompressible means, roughly, that a folded leaf can be straightened using an isotopy; there are no infinite folds. (II) Alternatively