Measured laminations in $3$-manifolds

Abstract:

An essential measured lamination embedded in an irreducible, orientable $3$-manifold $M$ is a codimension $1$ lamination with a transverse measure, carried by an incompressible branched surface satisfying further technical conditions. Weighted incompressible surfaces are examples of essential measured laminations, and the inclusion of a leaf of an essential measured lamination into $M$ is injective on ${\pi _1}$. There is a space $\mathcal {P}\mathcal {L}(M)$ whose points are projective classes of essential measured laminations. Projective classes of weighted incompressible surfaces are dense in $\mathcal {P}\mathcal {L}(M)$. The space $\mathcal {P}\mathcal {L}(M)$ is contained in a finite union of cells (of different dimensions) embedded in an infinite-dimensional projective space, and contains the interiors of these cells. Most of the properties of the incompressible branched surfaces carrying measured laminations are preserved under the operations of splitting or passing to sub-branched surfaces.