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”.

Classification : 51M10, 52B10, 57M50, 51F15
Keywords: Hyperbolic polyedra, dihedral angles, Andreev’s Theorem, Whitehead move, hyperbolic orbifold.
Mot clés : polyèdre hyperbolique, angles diédraux, théorème d’Andreev, déplacement de Withehea, orbite hyperbolique

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