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Geometric orbifolds.

William D. Dunbar — 1988

Revista Matemática de la Universidad Complutense de Madrid

An orbifold is a topological space which ?locally looks like? the orbit space of a properly discontinuous group action on a manifold. After a brief review of basic concepts, we consider the special case 3-dimensional orbifolds of the form GammaM, where M is a simply-connected 3-dimensional homogeneous space corresponding to one of Thurston?s eight geometries, and where Gamma < Isom(M) acts properly discontinuously. A general description of these geometric orbifolds is given and the closed...

Andreev’s Theorem on hyperbolic polyhedra

Roland K.W. Roeder;  John H. Hubbard;  William D. Dunbar — 2007

Annales de l’institut Fourier

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...

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