## Discrete Groups in Space and Uniformization ProblemsA revised and substantially enlarged edition of the Russian book Discrete transformation groups and manifold structures published by Nauka in 1983, this volume presents a comprehensive treatment of the geometric theory of discrete groups and the associated tessellations of the underlying space. Also |

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### Contents

Conformal mappings | 1 |

2 Quasiconformal mappings and isometry | 2 |

3 Möbius transformations | 4 |

4 Automorphisms of a ball and halfspace | 11 |

5 Fixed points of Möbius transformations | 15 |

6 Invariant domains for Möbius subgroups | 16 |

Geometrical structures on manifolds | 19 |

2 Homology and MayerVletoris sequences | 22 |

5 Invariant components for geometrically finite Kleinian groups | 154 |

6 The geometry of the Nielsen hull boundary | 168 |

Geometric and algebraic structures of discrete groups | 191 |

2 Free amalgamated products of Kleinian groups | 204 |

3 HNNextensions of Kleinian groups Examples | 212 |

4 Cayley graph and combinatorial structure of a discrete group | 223 |

5 Geometric and algebraic convergence of groups | 236 |

Sullivans ergodic theorems | 241 |

3 Spaces of constant curvature | 26 |

4 Representation of the Möbius group in the matrix group | 32 |

5 𝒱structures in manifolds | 34 |

6 Developing map and holonomy | 41 |

7 Orbifords and their 𝒱structures | 43 |

General properties of discontinuous groups | 51 |

2 Action of a group on the discontinuity set | 54 |

3 Some classes of Kleinian groups | 61 |

4 Fundamental domains | 64 |

the nonmeasurability effect and density property | 73 |

Discrete groups of hyperbolic Isometries | 77 |

2 Arithmetic construction of cocompact hyperbolic groups | 85 |

3 Universal bounds of hyperbolic manifold sizes | 87 |

4 Margulis lemma and hyperbolic manifold decomposition | 93 |

5 Precisely invariant horoballs and parabolic 𝜴peaks | 101 |

6 Crystallographic reflection groups and arithmetic | 108 |

Geometrically finite discrete Möbius groups | 117 |

2 The convex Nielsen hull of the limit set | 122 |

3 The limit set of a geometrically finite group | 126 |

4 Finiteness theorems for discrete Möbius groups | 141 |

Kleinian manifolds | 251 |

2 Topological aspects of combination theorems | 259 |

3 Kleinian manifold of a function group | 263 |

4 Isomorphism theorems | 283 |

5 Geometrically tame ends Kleinian manifold compactification and the Ahlfors problem on limitset measure | 290 |

Uniformization of manifolds | 307 |

2 Gromovs and Thurstons geometric uniformization concepts | 310 |

3 Thurstons hyperbolization theorem and the volumes of hyperbolic manifolds | 318 |

4 Complexity of 3manifolds integrable Hamiltonian systems and hyperbolic volumes | 328 |

5 Conformal uniformization of manifolds | 333 |

6 Conformal uniformization of doubles of atoroidal manifolds | 344 |

Deformation spaces of groups and structures | 355 |

2 Rigidity theorems for complete hyperbolic structures and isometry groups | 365 |

3 Deformation space for a Fuchsian group | 381 |

4 The boundary of deformation space for a Fuchsian group | 409 |

5 Deformations of Kleinian groups and conformal structures | 415 |

References | 425 |

467 | |

### Common terms and phrases

3-manifold algebraic Apanasov automorphisms ball boundary centre Chapter closed co-compact compact complement complete hyperbolic completes the proof condition conformal mapping conformal structure conjugate connected components consider construction contains converges convex Corollary corresponding covering cusp domain defined definition deformation denote dihedral angles disc discontinuity set discrete group embedding equivalent Euclidean example exists finite number fixed points Fuchsian group fundamental group fundamental polyhedron G-invariant geometrically finite group group G group GcMob(n half-space Hath holonomy homeomorphism homeomorphlsm homotopy horoballs horosphere hyperbolic manifolds hyperbolic space hyperbolic space Hn hyperbolic structures implies infinite intersect invariant components isometry group isomorphism Kleinian group Lemma limit set A(G loxodromic loxodromic elements manifold H metric Mob(n Mobius n-dimensional neighbourhood non-trivial obtained orbifold orthogonal parabolic cusps parabolic elements plane polyhedra proof of Theorem quasi-conformal radius representation sequence shows subgroup submanifold subset surface tessellation Thurston topological torus totally geodesic transformation vector xeRn