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Matings and the other side of the dictionary

John Hubbard — 2012

Annales de la faculté des sciences de Toulouse Mathématiques

In the theory of rational maps an important role is played by . These are probably the best understood of all rational functions, but they are bizarre, and involve gluing dendrites together to get spheres carrying Peano curves. In the theory of Kleinian groups, there is a parallel construction, the construction of , that is central to Thurston’s hyperbolization theorem for 3-manifolds that fiber over the circle with pseudo-Anosov monodromy. It also involves gluing dendrites and Peano curves. Clearly...

Andreev’s Theorem on hyperbolic polyhedra

Roland K.W. RoederJohn H. HubbardWilliam 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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