Hyperideal polyhedra in hyperbolic 3-space
Bulletin de la Société Mathématique de France, Volume 130 (2002) no. 3, pp. 457-491.

A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic 3-space 3 which, in the projective model for 3 ℝℙ 3 , is just the intersection of 3 with a projective polyhedron whose vertices are all outside 3 and whose edges all meet 3 . We classify hyperideal polyhedra, up to isometries of 3 , in terms of their combinatorial type and of their dihedral angles.

Un polyèdre hyperidéal est un polyèdre non-compact de l’espace hyperbolique 3 de dimension 3 qui, dans le modèle projectif pour 3 ℝℙ 3 , est simplement l’intersection de 3 avec un polyèdre projectif dont les sommets sont tous en dehors de 3 et dont toutes les arêtes rencontrent 3 . Nous classifions ces polyèdres hyperidéaux, à isométrie de 3 près, en fonction de leur type combinatoire et de leurs angles diédraux.

DOI: 10.24033/bsmf.2426
Classification: 51M09
Keywords: hyperbolic space, polyhedron, ideal polyhedron, hyperideal
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Bao, Xiliang; Bonahon, Francis. Hyperideal polyhedra in hyperbolic 3-space. Bulletin de la Société Mathématique de France, Volume 130 (2002) no. 3, pp. 457-491. doi : 10.24033/bsmf.2426. http://archive.numdam.org/articles/10.24033/bsmf.2426/

[1] A. Alexandrow - Konvexe Polyeder, Akademie-Verlag, Berlin, 1958. | MR

[2] E. Andreev - « On Convex Polyhedra in Lobačevskiĭ Spaces (Russian) », Mat. Sbornik 81 (123) (1970), p. 445-478; English transl., Math. USSR Sb. t.10 (1970), pp.413-440. | MR | Zbl

[3] -, « On Convex Polyhedra of Finite Volume in Lobačevskiĭ Spaces (Russian) », Mat. Sbornik 83 (125) (1970), p. 256-260; English transl., Math. USSR Sb. t.12 (1970), pp.255-259. | MR | Zbl

[4] X. Bao - Hyperideal Polyhedra in Hyperbolic 3-Space, Doctoral dissertation, University of Southern California, Los Angeles, 1998. | MR | Zbl

[5] A. Cauchy - « Sur les polygones et les polyèdres », J. École Polytechnique 9 (1813), p. 87-98.

[6] H. Coxeter - « On Complexes with Transitive Groups of Automorphisms », Ann. of Math. 35 (1934), p. 588-621. | JFM | Zbl

[7] P. Cromwell - Polyhedra, Cambridge University Press, 1997. | MR | Zbl

[8] B. Grünbaum - Convex Polytopes, Interscience Publishers, John Wiley & Sons Inc., New York, 1967. | MR | Zbl

[9] J. Van Lint & R. Wilson - A Course in Combinatorics, Cambridge University Press, 1992. | MR | Zbl

[10] I. Rivin - « Euclidean Structures on Simplicial Surfaces and Hyperbolic Volume », Ann. of Math. 139 (1994), p. 553-580. | MR | Zbl

[11] -, « A Characterization of Ideal Polyhedra in Hyperbolic 3-Space », Ann. of Math. 143 (1996), p. 51-70. | MR | Zbl

[12] I. Rivin & C. Hodgson - « A Characterization of Compact Convex Polyhedra in Hyperbolic 3-Space », Invent. Math. 111 (1993), p. 77-111; Corrigendum, Invent. Math. t.117 (1994), pp.359. | EuDML | MR | Zbl

[13] J.-M. Schlenker - « Métriques sur les polyèdres hyperboliques convexes », J. Diff. Geom. 48 (1998), p. 323-405. | MR | Zbl

[14] -, « Dihedral Angles of Convex Polyhedra », Discrete Comp. Geom. 23 (2000), p. 409-417. | MR | Zbl

[15] J. Stoker - « Geometrical Problems Concerning Polyhedra in the Large », Comm. Pure Appl. Math. 21 (1958), p. 119-168. | MR | Zbl

[16] W. Thurston - Three-Dimensional Geometry and Topology, (S. Levy, ed.), Princeton Math. Series, vol. 35, Princeton University Press, 1997. | MR | Zbl

[17] È. Vinberg - « Discrete groups generated by reflections in Lobačevskiĭ spaces », Mat. Sbornik 72 (114) (1967), p. 471-488; English transl., Math. USSR Sb. t.1 (1967), pp.429-444; Correction, Mat. Sbornik t.73 (115) (1967), pp.303. | MR | Zbl

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