## Kleinian Groups and Uniformization in Examples and ProblemsAimed at researchers, graduate students and undergraduates alike, this book presents a unified exposition of all the main areas and methods of the theory of Kleinian groups and the theory of uniformization of manifolds. The past 20 years have seen a rejuvenation of the field, due to the development of powerful new methods in topology, the theory of functions of several complex variables, and the theory of quasiconformal mappings. Thus this new book should provide a valuable resource, listing the basic facts regarding Kleinian groups and serving as a general guide to the primary literature, particularly the Russian literature in the field. In addition, the book includes a large number of examples, problems, and unsolved problems, many of them presented for the first time. |

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### Contents

Uniformization of Riemann Surfaces and Manifolds | 27 |

Examples | 63 |

Problems and Exercises further development of | 129 |

CONCLUSION Some Unsolved Problems | 171 |

27 | 173 |

179 | |

189 | |

195 | |

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### Common terms and phrases

Apanasov b-groups circle closed Riemann surface closed surface compact complete hyperbolic conformal automorphisms conformal mapping conformal structure conformally equivalent conjugate constructed corresponding cyclic group defined degenerate groups Denote diffeomorphism discrete group disk dodecahedron Euclidean example exists factor subgroup fiber Figure finite type finitely generated Kleinian fixed points free abelian free product Fuchsian group function fundamental group fundamental polygon fundamental polyhedron fundamental region fundamental set G1 and G2 geometrically finite group G C Mn Gusevskii holomorphic homeomorphic homotopy hyperbolic manifold hyperbolic space hyperbolic structure infinite invariant component isometric isometric spheres isomorphic Kleinian group Let G limit set loop loxodromic manifold M(G Maskit Math metric n-dimensional nontrivial obtained parabolic elements planar plane polyhedron Problem quasi-Fuchsian group quasiconformal deformations quasiconformal mappings Schottky group set of discontinuity simply connected space T(p subgroup of G surface of genus surface of type Teichmüller space three-dimensional manifold Thurston 234 topological torsion-free torus transformations uniformization