Measured pleated laminations of hyperbolic manifolds of dimension 3. (Laminations mesurées de plissage des variétés hyperboliques de dimension 3.) (French) Zbl 1083.57023

For a hyperbolic metric on a compact 3-dimensional manifold \(M\), the boundary of its convex core \(C_m\) (\(m\) is the hyperbolic metric, non-fuchsian and geometrically finite) is a surface which is almost everywhere totally geodesic, but which is bent along a family of disjoint geodesics. The locus and the intensity of this bending are described by a measured geodesic lamination, which is a topological object. The measured pleated lamination of the boundary of \(C_m\) is an invariant of the hyperbolic metric \(m\) on \(M\). The space of measured geodesic laminations on the boundary \(\partial \overline M\) uniquely depends on the topology of \(\partial \overline M\). The authors consider a function from the space of geometrically finite metrics of \(M\) to the space of measured geodesic laminations on the boundary, which maps each hyperbolic metric to its measured pleated lamination and study the properties of this map. They consider two problems: the topological characterization of those measured geodesic laminations which can occur as bending measured laminations of hyperbolic metrics and the uniqueness problem which asks whether a hyperbolic metric is uniquely determined by its bending measured lamination. The authors obtain a complete response to the first problem under the hypothesis that the boundary is incompressible.


57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R30 Foliations in differential topology; geometric theory
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