Degenerations of hyperbolic structures. II: Measured laminations in 3- manifolds.

*(English)*Zbl 0656.57003This paper is the second in a series of three papers. For the first part see Zbl 0583.57005. Despite the title, hyperbolic structures are not considered in this paper. It is devoted to the foundation of a general theory of measured laminations in 3-manifolds. In the third paper [J. Morgan and P. Shalen, Degenerations of hyperbolic structures. III: Actions of 3-manifold groups on trees and Thurston’s compactness theorem, Ann. Math., II. Ser. 127, No.3, 457-519 (1988)] the results of this paper are combined with the results of the first part to study degenerations of hyperbolic structures on 3-manifolds.

The authors introduce measured laminations in 3-manifolds as a generalization of two concepts simultaneously: geodesic laminations on surfaces and surfaces in 3-manifolds. According to the authors, “most of this paper consists in extending Thurston’s theory of geodesic laminations and the Haken-Stallings-Waldhausen theory of incompressible surfaces to our context”. A measured lamination in a manifold \({\mathcal M}\) is, by definition, a foliation of a closed subset of \({\mathcal M}\) with codimension 1 leaves (the leaves are smooth submanifolds) and provided with a transverse measure. The general theory of such objects is in many respects similar to the theory of foliations and is developed in Chapter I. A measured lamination can be constructed by assigning some positive weights to the sheets of a branched surface. This is quite similar to the Thurston’s theory of train tracks. This construction is studied in Chapter II. At the end of this chapter an important concept of flat branched surface is introduced. (By technical reasons, the authors use not branched surfaces but an equivalent notion of a branched I-bundle.) Chapter III is a generalization of Haken’s theory of normal surfaces in 3-manifolds. The final Chapter IV is devoted to incompressible branched surfaces. An important theorem asserts that the leaves of a lamination constructed from an incompressible branched surface are algebraically incompressible in the 3-manifold. Using this theorem the authors prove that any flat, incompressible branched surface in a Haken 3-manifold deforms into the (characteristic) graph submanifold. This theorem is of crucial importance for the third paper in the series. Some results of this paper overlap with recent work of U. Oertel, Measured laminations in 3-manifolds, Trans. Am. Math. Soc. 305, No.2, 531-573 (1988).

The authors introduce measured laminations in 3-manifolds as a generalization of two concepts simultaneously: geodesic laminations on surfaces and surfaces in 3-manifolds. According to the authors, “most of this paper consists in extending Thurston’s theory of geodesic laminations and the Haken-Stallings-Waldhausen theory of incompressible surfaces to our context”. A measured lamination in a manifold \({\mathcal M}\) is, by definition, a foliation of a closed subset of \({\mathcal M}\) with codimension 1 leaves (the leaves are smooth submanifolds) and provided with a transverse measure. The general theory of such objects is in many respects similar to the theory of foliations and is developed in Chapter I. A measured lamination can be constructed by assigning some positive weights to the sheets of a branched surface. This is quite similar to the Thurston’s theory of train tracks. This construction is studied in Chapter II. At the end of this chapter an important concept of flat branched surface is introduced. (By technical reasons, the authors use not branched surfaces but an equivalent notion of a branched I-bundle.) Chapter III is a generalization of Haken’s theory of normal surfaces in 3-manifolds. The final Chapter IV is devoted to incompressible branched surfaces. An important theorem asserts that the leaves of a lamination constructed from an incompressible branched surface are algebraically incompressible in the 3-manifold. Using this theorem the authors prove that any flat, incompressible branched surface in a Haken 3-manifold deforms into the (characteristic) graph submanifold. This theorem is of crucial importance for the third paper in the series. Some results of this paper overlap with recent work of U. Oertel, Measured laminations in 3-manifolds, Trans. Am. Math. Soc. 305, No.2, 531-573 (1988).

Reviewer: N.V.Ivanov

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57R30 | Foliations in differential topology; geometric theory |

57R50 | Differential topological aspects of diffeomorphisms |