×

Andreev’s theorem on hyperbolic polyhedra. (English) Zbl 1127.51012

The authors correct and reconstruct the proof of the famous Andreev’s theorem, which shows a classfication of all 3-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles (except tetrahedra). Andreev’s theorem is very important in recent researches of the 3-dimensional hyperbolic geometry. This work needs lots of effort of considering a combinatorial problem and hyperbolic dihedral angles about classification.

MSC:

51M10 Hyperbolic and elliptic geometries (general) and generalizations
52B10 Three-dimensional polytopes
57M50 General geometric structures on low-dimensional manifolds
51F15 Reflection groups, reflection geometries
PDFBibTeX XMLCite
Full Text: DOI arXiv Numdam EuDML

References:

[1] Aigner, M.; Ziegler, G. M., Proofs from The Book, third ed. (2004) · Zbl 1038.00001
[2] Alekseevskij, D. V.; Vinberg, È. B.; Solodovnikov, A. S., Encyclopaedia Math. Sci., 29, 1-138 (1993) · Zbl 0787.53001
[3] Andreev, E. M., Convex polyhedra in Lobačevskiĭ spaces (english transl.), Math. USSR Sbornik, 10, 413-440 (1970) · Zbl 0217.46801 · doi:10.1070/SM1970v010n03ABEH001677
[4] Andreev, E. M., Convex polyhedra in Lobačevskiĭ spaces (in Russian), Mat. Sb., 81, 445-478 (1970) · Zbl 0194.23202
[5] Bao, Xiliang; Bonahon, Francis, Hyperideal polyhedra in hyperbolic 3-space, Bull. Soc. Math. France, 130, 457-491 (2002) · Zbl 1033.52009
[6] Boileau, Michel, Uniformisation en dimension trois, Séminaire Bourbaki 1998/99, exposé 855, Astérisque, 266, 137-174 (2000) · Zbl 0942.57013
[7] Boileau, Michel; Porti, Joan, Geometrization of 3-orbifolds of cyclic type, 272 (2001) · 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, 185-215 (1996) · Zbl 0856.51012 · doi:10.1112/plms/s3-73.1.185
[9] Chow, Bennett; Luo, Feng, Combinatorial Ricci flows on surfaces, J. Diff. Geom., 63, 97-129 (2003) · Zbl 1070.53040
[10] Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P., MSJ Memoirs, 5 (2000) · 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, 993-998 (1997) · Zbl 0898.52010 · doi:10.1016/S0764-4442(97)89092-1
[13] Díaz, Raquel, A generalization of Andreev’s theorem, J. Math. Soc. Japan, 58, 333-349 (2006) · Zbl 1097.51009 · doi:10.2969/jmsj/1149166778
[14] Douady, Régine; Douady, Adrien, Algèbre et théories galoisiennes, 2 (1979) · 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, 111-124 (2004) · Zbl 1065.52008 · doi:10.1007/s10711-004-3180-y
[16] Hodgson, C. D., Deduction of Andreev’s theorem from Rivin’s characterization of convex hyperbolic polyhedra, Topology, 90, 185-193 (1992) · Zbl 0765.52013
[17] Kapovich, Michael, Progress in Math., 183 (2001) · Zbl 0958.57001
[18] Lima, Elon Lages, Fundamental groups and covering spaces (translated from Portuguese by Jonas Gomes) (2003) · Zbl 1029.55001
[19] Marden, A.; Rodin, B., Lecture Notes in Math., 1435, 103-115 (1990) · Zbl 0717.52014
[20] Otal, Jean-Pierre, Surveys in Differential Geometry, Cambridge, MA, 1996, III, 77-194 (1998) · Zbl 0997.57001
[21] Rivin, I.; Hodgson, C. D., A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math., 111, 77-111 (1993) · Zbl 0784.52013 · doi:10.1007/BF01231281
[22] Rivin, Igor, On geometry of convex ideal polyhedra in hyperbolic \(3\)-space, Topology, 32, 87-92 (1993) · Zbl 0784.52014 · doi:10.1016/0040-9383(93)90039-X
[23] Rivin, Igor, A characterization of ideal polyhedra in hyperbolic \(3\)-space, Ann. of Math. (2), 143, 51-70 (1996) · Zbl 0874.52006 · doi:10.2307/2118652
[24] Rivin, Igor, Combinatorial optimization in geometry, Adv. Appl. Math., 31, 242-271 (2003) · Zbl 1028.52006 · doi:10.1016/S0196-8858(03)00093-9
[25] Roeder, Roland K. W., Compact hyperbolic tetrahedra with non-obtuse dihedral angles, Publications Mathématiques, 50, 211-227 (2006) · Zbl 1127.52010
[26] Roeder, Roland K. W., Le théorème d’Andreev sur polyèdres hyperboliques (in English) (2004)
[27] Schlenker, J.-M., Dihedral angles of convex polyhedra, Discrete Comput. Geom., 23, 409-417 (2000) · Zbl 0951.52006 · doi:10.1007/PL00009509
[28] Schlenker, Jean-Marc, Métriques sur les polyèdres hyperboliques convexes, J. Differential Geom., 48, 323-405 (1998) · Zbl 0912.52008
[29] Schlenker, Jean-Marc, Hyperbolic manifolds with convex boundary, Invent. Math., 163, 109-169 (2006) · Zbl 1091.53019 · doi:10.1007/s00222-005-0456-x
[30] Thurston, W. P., Geometry and topology of 3-manifolds (19781979)
[31] Thurston, William P., Princeton Mathematical Series, 35 (1997) · Zbl 0873.57001
[32] Vinberg, È. B., Discrete groups generated by reflections in Lobačevskiĭ spaces, Mat. Sb. (N.S.), 72, 471-488 (1967) · Zbl 0166.16303
[33] Vinberg, È. B., Hyperbolic groups of reflections, Russian Math. Surveys, 40, 31-75 (1985) · Zbl 0579.51015 · doi:10.1070/RM1985v040n01ABEH003527
[34] Vinberg, È. B., Amer. Math. Soc. Transl. Ser. 2, 148, 15-27 (1991) · Zbl 0742.51019
[35] Vinberg, È. B.; Shvartsman, O. V., Encyclopaedia Math. Sci., 29, 139-248 (1993) · Zbl 0787.22012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.