Andreev’s Theorem on hyperbolic polyhedra

Roland K.W. Roeder[1]; John H. Hubbard[2]; William D. Dunbar[3]

  • [1] Fields Institute 222 College St. Toronto ON M5T 3J1 (Canada)
  • [2] Cornell University Mallot Hall Ithaca, NY 14853 (USA) and Université de Provence Centre de Mathématiques et d’Informatique 39 rue Joliot-Curie 13453 Marseille Cedex 13 (France)
  • [3] Simon’s Rock College of Bard 84 Alford Road Great Barrington, MA 01230 (USA)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 3, page 825-882
  • ISSN: 0373-0956

Abstract

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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”.

How to cite

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Roeder, Roland K.W., Hubbard, John H., and Dunbar, William D.. "Andreev’s Theorem on hyperbolic polyhedra." Annales de l’institut Fourier 57.3 (2007): 825-882. <http://eudml.org/doc/10244>.

@article{Roeder2007,
abstract = {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”.},
affiliation = {Fields Institute 222 College St. Toronto ON M5T 3J1 (Canada); Cornell University Mallot Hall Ithaca, NY 14853 (USA) and Université de Provence Centre de Mathématiques et d’Informatique 39 rue Joliot-Curie 13453 Marseille Cedex 13 (France); Simon’s Rock College of Bard 84 Alford Road Great Barrington, MA 01230 (USA)},
author = {Roeder, Roland K.W., Hubbard, John H., Dunbar, William D.},
journal = {Annales de l’institut Fourier},
keywords = {Hyperbolic polyedra; dihedral angles; Andreev’s Theorem; Whitehead move; hyperbolic orbifold; hyperbolic polyhedra; dihederal angle; Andreev's Theorem},
language = {eng},
number = {3},
pages = {825-882},
publisher = {Association des Annales de l’institut Fourier},
title = {Andreev’s Theorem on hyperbolic polyhedra},
url = {http://eudml.org/doc/10244},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Roeder, Roland K.W.
AU - Hubbard, John H.
AU - Dunbar, William D.
TI - Andreev’s Theorem on hyperbolic polyhedra
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 3
SP - 825
EP - 882
AB - 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”.
LA - eng
KW - Hyperbolic polyedra; dihedral angles; Andreev’s Theorem; Whitehead move; hyperbolic orbifold; hyperbolic polyhedra; dihederal angle; Andreev's Theorem
UR - http://eudml.org/doc/10244
ER -

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