Title:
Combinatorial Aspects of Andreev's Classification of Hyperbolic Polyhedra
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Abstract:
In 1970, E. M. Andreev published a classification of all
three-dimensional compact hyperbolic polyhedra 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. This 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. It is also remarkable to what level the
proof of Andreev's Theorem resembles (in a simpler way) the proof of
Thurston.