Degenerations of Hyperbolic Structures, II: Measured Laminations in 3-Manifolds

That paper concerned the general theory of groups acting on R-trees and the relationship of these actions to representations into SL2(C). The purpose of the present paper is to develop the foundations of a general theory of measured laminations in 3-manifolds. In the third paper [8] the results of this paper will be applied to study actions of 3-manifold groups on trees. (The starting point of [8] is the idea that dual to an action of qg,(M) on an R-tree is a codimension-1 measured lamination in M.) Combining the latter results with [7] gives information about how hyperbolic structures on 3-manifolds can degenerate. We view measured laminations in 3-manifolds as generalizations of both geodesic laminations on surfaces and surfaces in 3-manifolds. Most of this paper consists in extending Thurston's theory of geodesic laminations and the HakenStallings-Waldhausen theory of incompressible surfaces to our context. The paper is organized along the following lines. In Chapter I we develop the basic notions of codimension-1 measured laminations. We prove one new, and quite useful, result (Theorem I.3.2) concerning a decomposition of these objects. We also give a definition of the Euler characteristic of a measured lamination and show that it has all the usual properties. In Chapter II we introduce the branched surfaces of Williams [16] and Hatcher-Thurston. We establish analogues for measured laminations carried by branched surfaces of some results of Thurston's for train tracks and geodesic laminations. By imitating an argument of Plante's [10], we show that the leaves of a measured lamination have polynomial growth. This is used to show that if a branched surface N carries some lamination all of whose leaves have virtually abelian fundamental groups and carries no disks or spheres, then all laminations carried by N have zero Euler characteristic. We express the latter property by saying that N is flat. In Chapter III, generalizing Haken's notion of normal surfaces, we develop the theory of normal measured laminations and of normal branched surfaces in a