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Roeder, Roland K. W.; Hubbard, John H.; Dunbar, William D.
Andreev’s theorem on hyperbolic polyhedra. Annales de l'institut Fourier, 57 no. 3 (2007), p. 825-882
Texte intégral djvu | pdf | Analyses MR 2336832 | Zbl 1127.51012
Class. Math.: 51M10, 52B10, 57M50, 51F15
URL stable: http://www.numdam.org/item?id=AIF_2007__57_3_825_0
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E.M. Andreev a publié en 1970 une classification des polyèdres hyperboliques compacts de dimension 3 (autre que les tétraèdres) dont les angles dièdres sont non-obtus. Étant donné une description combinatoire d’un polyèdre $C$, le théorème d’Andreev dit que les angles dièdres possibles sont exactement décrits par cinq classes d’inégalités linéaires. Le théorème d’Andreev démontre également que le polyèdre résultant est alors unique à isométrie hyperbolique près.
D’une part, le théorème d’Andreev est évidemment un énoncé intéressant de la géométrie de l’espace hyperbolique en dimension 3 ; d’autre part c’est un outil essentiel dans la preuve du théorème d’hyperbolisation de Thurston pour les variétés Haken de dimension 3.
La démonstration d’Andreev contient une erreur importante. Nous corrigeons ici cette erreur et nous fournissons aussi une nouvelle preuve lisible des autres parties de la preuve, car l’article d’Andreev a la réputation d’être «illisible».
[1] Aigner M. & Ziegler G. M., Proofs from The Book, third ed., Springer-Verlag, Berlin, 2004 MR 2014872 | Zbl 1038.00001
[2] Alekseevskij D. V., Vinberg È. B. & Solodovnikov A. S., Geometry of spaces of constant curvature, Geometry, II, Encyclopaedia Math. Sci., 29, Springer, Berlin, 1993 MR 1254932 | Zbl 0787.53001
[3] Andreev E. M., Convex polyhedra in Lobačevskiĭ spaces (english transl.), Math. USSR Sbornik 10 (1970) p. 413-440 MR 259734 | Zbl 0217.46801
[4] Andreev E. M., Convex polyhedra in Lobačevskiĭ spaces (in Russian), Mat. Sb. 81 (1970) p. 445-478 MR 259734 | Zbl 0194.23202
[5] Bao Xiliang & Bonahon Francis, Hyperideal polyhedra in hyperbolic 3-space, Bull. Soc. Math. France 130 (2002) p. 457-491 MR 1943885 | Zbl 1033.52009
[6] Boileau Michel, Uniformisation en dimension trois, Séminaire Bourbaki 1998/99, exposé 855, Astérisque 266 (2000) p. 137-174
Numdam | MR 1772673 | Zbl 0942.57013
[7] Boileau Michel & Porti Joan, Geometrization of 3-orbifolds of cyclic type, 272, SMF, 2001 MR 1844891 | Zbl 0971.57004
[8] Bowers P. & Stephenson K., A branched Andreev-Thurston theorem for circle packings of the sphere, Proc. London Math. Soc. (3) 73 (1996) p. 185-215 MR 1387087 | Zbl 0856.51012
[9] Chow Bennett & Luo Feng, Combinatorial Ricci flows on surfaces, J. Diff. Geom. 63 (2003) p. 97-129
Article | MR 2015261 | Zbl 1070.53040
[10] Cooper Daryl, Hodgson Craig D. & Kerckhoff Steven P., Three-dimensional orbifolds and cone-manifolds, with a postface by Sadayoshi Kojima, MSJ Memoirs, 5, Math. Society of Japan, Tokyo, 2000 MR 1778789 | Zbl 0955.57014
[11] Developed by The Geometry Center at the University of Minnesota in the late 1990’s , www.geomview.org
[12] Díaz Raquel, Non-convexity of the space of dihedral angles of hyperbolic polyhedra, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) p. 993-998 MR 1485617 | Zbl 0898.52010
[13] Díaz Raquel, A generalization of Andreev’s theorem, J. Math. Soc. Japan 58 (2006) p. 333-349
Article | MR 2228562 | Zbl 1097.51009
[14] Douady Régine & Douady Adrien, Algèbre et théories galoisiennes, 2, CEDIC, Paris, 1979 MR 595328 | Zbl 1076.12004
[15] Guéritaud François, On an elementary proof of Rivin’s characterization of convex ideal hyperbolic polyhedra by their dihedral angles, Geom. Dedicata 108 (2004) p. 111-124 MR 2112668 | Zbl 1065.52008
[16] Hodgson C. D., Deduction of Andreev’s theorem from Rivin’s characterization of convex hyperbolic polyhedra, Topology 90 (1992) p. 185-193 MR 1184410 | Zbl 0765.52013
[17] Kapovich Michael, Hyperbolic manifolds and discrete groups, Progress in Math., 183, Birkhäuser Boston, 2001 MR 1792613 | Zbl 0958.57001
[18] Lima Elon Lages, Fundamental groups and covering spaces (translated from Portuguese by Jonas Gomes), AK Peters Ltd., Natick, MA, 2003 MR 2000701 | Zbl 1029.55001
[19] Marden A. & Rodin B., On Thurston’s formulation and proof of Andreev’s Theorem, in Computational Methods and Function Theory, Lecture Notes in Math., 1435, Springer-Verlag, 1990 MR 1071766 | Zbl 0717.52014
[20] Otal Jean-Pierre, Thurston’s hyperbolization of Haken manifolds, Surveys in Differential Geometry, Cambridge, MA, 1996, III, Int. Press, 1998 MR 1677888 | Zbl 0997.57001
[21] Rivin I. & Hodgson C. D., A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math. 111 (1993) p. 77-111 MR 1193599 | Zbl 0784.52013
[22] Rivin Igor, On geometry of convex ideal polyhedra in hyperbolic $3$-space, Topology 32 (1993) p. 87-92 MR 1204408 | Zbl 0784.52014
[23] Rivin Igor, A characterization of ideal polyhedra in hyperbolic $3$-space, Ann. of Math. (2) 143 (1996) p. 51-70 MR 1370757 | Zbl 0874.52006
[24] Rivin Igor, Combinatorial optimization in geometry, Adv. Appl. Math. 31 (2003) p. 242-271 MR 1985831 | Zbl 1028.52006
[25] Roeder Roland K. W., Compact hyperbolic tetrahedra with non-obtuse dihedral angles, Publications Mathématiques 50 (2006) p. 211-227 Zbl 1127.52010
[26] Roeder Roland K. W., Le théorème d’Andreev sur polyèdres hyperboliques (in English), Ph. D. Thesis, Université de Provence, Aix-Marseille 1, May 2004
[27] Schlenker J.-M., Dihedral angles of convex polyhedra, Discrete Comput. Geom. 23 (2000) p. 409-417 MR 1744513 | Zbl 0951.52006
[28] Schlenker Jean-Marc, Métriques sur les polyèdres hyperboliques convexes, J. Differential Geom. 48 (1998) p. 323-405 MR 1630178 | Zbl 0912.52008
[29] Schlenker Jean-Marc, Hyperbolic manifolds with convex boundary, Invent. Math. 163 (2006) p. 109-169 MR 2208419 | Zbl 05000067
[30] Thurston W. P., Geometry and topology of 3-manifolds, Princeton University Lecture Notes, 1978-1979
[31] Thurston William P., Three-dimensional geometry and topology, 1, Princeton Mathematical Series, 35, Princeton University Press, 1997 MR 1435975 | Zbl 0873.57001
[32] Vinberg È. B., Discrete groups generated by reflections in Lobačevskiĭ spaces, Mat. Sb. (N.S.) 72 (1967) p. 471-488 MR 207853 | Zbl 0166.16303
[33] Vinberg È. B., Hyperbolic groups of reflections, Russian Math. Surveys 40 (1985) p. 31-75 MR 783604 | Zbl 0579.51015
[34] Vinberg È. B., The volume of polyhedra on a sphere and in Lobachevsky space, Algebra and analysis (Kemerovo, 1988), Amer. Math. Soc. Transl. Ser. 2, 148, Amer. Math. Soc., Providence, RI, 1991 Zbl 0742.51019
[35] Vinberg È. B. & Shvartsman O. V., Discrete groups of motions of spaces of constant curvature, in Geometry, II, Encyclopaedia Math. Sci., 29, Springer, Berlin, 1993 MR 1254933 | Zbl 0787.22012
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