Measured Lamination Spaces for 3-Manifolds

One of Thurston’s important contributions to surface theory is his construction of measured lamination spaces. These spaces originally arose because their projectivizations form natural boundaries for Teichmüller spaces, but they are also of interest purely topologically. For example, they provide a nice global framework in which to view all the isotopy classes of simple closed curves in a surface. In view of the many parallels between surface theory and 3 manifold theory, it is natural to ask whether 3 manifolds also have measured lamination spaces. The idea would be that, after projectivization, the rational points of the measured lamination space of a given 3 manifold M would be the isotopy classes of incompressible surfaces in M , and the remaining points would be isotopy classes of measured laminations in M satisfying some sort of incompressibility conditions. There is a 1988 paper of Oertel [O1] which takes some first steps in this direction. At about the same time I also put some effort into a project of working out the technical details of this theory. Unfortunately these details turned out to be much more complicated than I would have liked, and there were no striking applications on the horizon, so the project was eventually abandoned. Still, there are some nice ideas here which may some day prove useful, so it may be worthwhile to make some of this material available, even without full proofs of all the stated results.