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About the rigidity of tridimensional hyperbolic polyhedra: finite volume case, hyperideal case, Fuchsian case. (Sur la rigidité de polyèdres hyperboliques en dimension \(3\): cas de volume fini, cas hyperidéal, cas fuchsien.) (French) Zbl 1061.52007

The article contains the following principal results as well as their generalisation to the case of Fuchsian polyhedra:
(1) A characterisation of the hyperbolic convex polyhedra with \(n\) semi-ideal faces (with vertices at infinity) in terms of their dual metric. As particular cases, that characterisation comprises the case of compact polyhedra previously considered by E. M. Andreev [Math. USSR, Sb. 10, 413–440 (1970; Zbl 0217.46801)] and the case of ideal polyhedra previously considered by I. Rivin and C. D. Hodgson [Invent. Math. 111, No. 1, 77–111 (1993; Zbl 0784.52013)] and by I. Rivin [Ann. Math. (2) 143, No. 1, 51–70 (1996; Zbl 0874.52006)].
(2) A new proof of the characterisation of the hyperbolic hyperideal convex polyhedra of a prescribed combinatorial type (the vertices are at infinity or behind the absolute) by their dihedral angles.

MSC:

52B10 Three-dimensional polytopes
52A55 Spherical and hyperbolic convexity
51M10 Hyperbolic and elliptic geometries (general) and generalizations
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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