Andreev’s Theorem on hyperbolic polyhedra

Roeder, Roland K.W.; Hubbard, John H.; Dunbar, William D.

Annales de l'Institut Fourier, Tome 57 (2007) no. 3, pp. 825-882.

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”.

In 1970, E.M.Andreev published a classification of all three-dimensional compact hyperbolic polyhedra (other than tetrahedra) having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev’s Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing $C$ with the assigned dihedral angles. Andreev’s Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry.

Andreev’s Theorem is both an interesting statement about the geometry of hyperbolic 3-dimensional space, as well as a fundamental tool used in the proof for Thurston’s Hyperbolization Theorem for 3-dimensional Haken manifolds.

We correct a fundamental error in Andreev’s proof of existence and also provide a readable new proof of the other parts of the proof of Andreev’s Theorem, because Andreev’s paper has the reputation of being “unreadable”.

[1] Aigner, M.; Ziegler, G. M. Proofs from The Book, third ed., Springer-Verlag, Berlin, 2004 | MR | Zbl

[2] Alekseevskij, D. V.; Vinberg, È. B.; Solodovnikov, A. S. Geometry of spaces of constant curvature, Geometry, II, Encyclopaedia Math. Sci., Volume 29, Springer, Berlin, 1993, pp. 1-138 | MR | Zbl

[3] Andreev, E. M. Convex polyhedra in Lobačevskiĭ spaces (english transl.), Math. USSR Sbornik, Volume 10 (1970), pp. 413-440 | DOI | MR | Zbl

[4] Andreev, E. M. Convex polyhedra in Lobačevskiĭ spaces (in Russian), Mat. Sb., Volume 81 (1970), pp. 445-478 | MR | Zbl

[5] Bao, Xiliang; Bonahon, Francis Hyperideal polyhedra in hyperbolic 3-space, Bull. Soc. Math. France, Volume 130 (2002), pp. 457-491 | Numdam | MR | Zbl

[6] Boileau, Michel Uniformisation en dimension trois, Séminaire Bourbaki 1998/99, exposé 855, Astérisque, Volume 266 (2000), pp. 137-174 | Numdam | MR | Zbl

[7] Boileau, Michel; Porti, Joan Geometrization of 3-orbifolds of cyclic type, 272, SMF, 2001 (Appendix A by Michael Heusener and Porti) | Numdam | MR | Zbl

[8] Bowers, P.; Stephenson, K. A branched Andreev-Thurston theorem for circle packings of the sphere, Proc. London Math. Soc. (3), Volume 73 (1996), pp. 185-215 | DOI | MR | Zbl

[9] Chow, Bennett; Luo, Feng Combinatorial Ricci flows on surfaces, J. Diff. Geom., Volume 63 (2003), pp. 97-129 | MR | Zbl

[10] Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P. Three-dimensional orbifolds and cone-manifolds, with a postface by Sadayoshi Kojima, MSJ Memoirs, Volume 5, Math. Society of Japan, Tokyo, 2000 | MR | Zbl

[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., Volume 325 (1997), pp. 993-998 | DOI | MR | Zbl

[13] Díaz, Raquel A generalization of Andreev’s theorem, J. Math. Soc. Japan, Volume 58 (2006), pp. 333-349 | DOI | MR | Zbl

[14] Douady, Régine; Douady, Adrien Algèbre et théories galoisiennes, 2, CEDIC, Paris, 1979 | MR | Zbl

[15] Guéritaud, François On an elementary proof of Rivin’s characterization of convex ideal hyperbolic polyhedra by their dihedral angles, Geom. Dedicata, Volume 108 (2004), pp. 111-124 | DOI | MR | Zbl

[16] Hodgson, C. D. Deduction of Andreev’s theorem from Rivin’s characterization of convex hyperbolic polyhedra, Topology, Volume 90 (1992), pp. 185-193 | MR | Zbl

[17] Kapovich, Michael Hyperbolic manifolds and discrete groups, Progress in Math., Volume 183, Birkhäuser Boston, 2001 | MR | Zbl

[18] Lima, Elon Lages Fundamental groups and covering spaces (translated from Portuguese by Jonas Gomes), AK Peters Ltd., Natick, MA, 2003 | MR | Zbl

[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., Volume 1435, Springer-Verlag, 1990, pp. 103-115 | MR | Zbl

[20] Otal, Jean-Pierre Thurston’s hyperbolization of Haken manifolds, Surveys in Differential Geometry, Cambridge, MA, 1996, Volume III, Int. Press, 1998, pp. 77-194 | MR | Zbl

[21] Rivin, I.; Hodgson, C. D. A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math., Volume 111 (1993), pp. 77-111 | DOI | MR | Zbl

[22] Rivin, Igor On geometry of convex ideal polyhedra in hyperbolic $3$-space, Topology, Volume 32 (1993), pp. 87-92 | DOI | MR | Zbl

[23] Rivin, Igor A characterization of ideal polyhedra in hyperbolic $3$-space, Ann. of Math. (2), Volume 143 (1996), pp. 51-70 | DOI | MR | Zbl

[24] Rivin, Igor Combinatorial optimization in geometry, Adv. Appl. Math., Volume 31 (2003), pp. 242-271 | DOI | MR | Zbl

[25] Roeder, Roland K. W. Compact hyperbolic tetrahedra with non-obtuse dihedral angles, Publications Mathématiques, Volume 50 (2006), pp. 211-227 | Zbl

[26] Roeder, Roland K. W. Le théorème d’Andreev sur polyèdres hyperboliques (in English), Université de Provence, Aix-Marseille 1 (May 2004) (Ph. D. Thesis)

[27] Schlenker, J.-M. Dihedral angles of convex polyhedra, Discrete Comput. Geom., Volume 23 (2000), pp. 409-417 | DOI | MR | Zbl

[28] Schlenker, Jean-Marc Métriques sur les polyèdres hyperboliques convexes, J. Differential Geom., Volume 48 (1998), pp. 323-405 | MR | Zbl

[29] Schlenker, Jean-Marc Hyperbolic manifolds with convex boundary, Invent. Math., Volume 163 (2006), pp. 109-169 | DOI | MR | Zbl

[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, Volume 35, Princeton University Press, 1997 | MR | Zbl

[32] Vinberg, È. B. Discrete groups generated by reflections in Lobačevskiĭ spaces, Mat. Sb. (N.S.), Volume 72 (1967), pp. 471-488 correction, ibid. 73 (115) (1967), 303 | MR | Zbl

[33] Vinberg, È. B. Hyperbolic groups of reflections, Russian Math. Surveys, Volume 40 (1985), pp. 31-75 | DOI | MR | Zbl

[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, Volume 148, Amer. Math. Soc., Providence, RI, 1991, pp. 15-27 | Zbl

[35] Vinberg, È. B.; Shvartsman, O. V. Discrete groups of motions of spaces of constant curvature, in Geometry, II, Encyclopaedia Math. Sci., Volume 29, Springer, Berlin, 1993, pp. 139-248 | MR | Zbl