## 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

1 | |

Uniformization of Riemann Surfaces and Manifolds | 27 |

Examples | 63 |

Problems and Exercises further development of the theory | 129 |

Some Unsolved Problems | 171 |

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### Common terms and phrases

Apanasov boundary circle closed Riemann surface compact complete hyperbolic conformal automorphisms conformal mapping conformal structure conformally equivalent conjugate consider constructed contains corresponding curve cyclic group defined degenerate groups Dehn surgery Denote diffeomorphism discrete group disk dodecahedron Euclidean example exists factor subgroup fi(G 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 geometrically finite group G C Mn holomorphic homeomorphic homotopy hyperbolic manifold hyperbolic structure infinite invariant component isometric isometric spheres isomorphic Kleinian group knot Let G loops loxodromic manifold M(G Maskit Math metric Mobius mappings n-dimensional nontrivial obtained parabolic elements plane Poincare Problem Prove quasi-Fuchsian group quasicircles quasiconformal automorphism quasiconformal deformations quasiconformal mappings Schottky group set of discontinuity simply connected subgroup of G surface of genus Teichmiiller space three-dimensional manifold topological torsion-free torus transformations trivial uniformization

### Popular passages

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Page 184 - Lobacevskir space groups with an infinite fundamental region of finite measure. Dokl. Akad. Nauk SSSR 167 (1966), 30-33 = Soviet Math.

Page 179 - H. Garland and MS Raghunathan, Fundamental domains for lattices in (R-) rank 1 semisimple Lie groups, Ann. of Math. (2) 92 (1970), 279-326.

Page 183 - Geometrie der konformen Abbildung auf der hyperbolischen und der elliptischen Ebene, VEB Deutscher Verlag Wiss., Berlin, 1974.

Page 187 - Stallings, A topological proof of Grushko's theorem on free products, Math. Z. 90 (1965) 1-8.

Page 179 - Univalent functions and the Schwarzian derivative. Comment. Math. Helv. 52 (1977), 561-572.