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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.

##### MSC:
 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
##### Keywords:
3-manifold; hyperbolic metric
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