Octahedral Noncompact Hyperbolic Space Forms With Finite Volume

Marica Šarac

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In this note we construct examples of geometric 3-orbifolds with (orbifold) fundamental group isomorphic to a (Z-extension of a) generalized Coxeter group. Some of these orbifolds have either euclidean, spherical or hyperbolic structure. As an application, we obtain an alternative proof of theorem 1 of Hagelberg, Maclaughlan and Rosenberg in [5]. We also obtain a similar result for generalized Coxeter groups.

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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...

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Quello delle triangolazioni geodetiche ideali è un metodo molto potente per costruire strutture iperboliche complete di volume finito su 3-varietà non compatte, ma non è noto se il metodo sia applicabile in generale. È tuttavia noto che esistono triangolazioni ideali parzialmente piatte, ma l'analisi della situazione diviene più ardua sotto diversi aspetti, quando si ha a che fare con tetraedri piatti oltre che veri tetraedri. In particolare, la topologia dello spazio di identificazione...