## Three-dimensional Geometry and Topology, Volume 1This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty. This book was the origin of a grand scheme developed by Thurston that is now coming to fruition. In the 1920s and 1930s the mathematics of two-dimensional spaces was formalized. It was Thurston's goal to do the same for three-dimensional spaces. To do this, he had to establish the strong connection of geometry to topology--the study of qualitative questions about geometrical structures. The author created a new set of concepts, and the expression "Thurston-type geometry" has become a commonplace.
In 2005 Thurston won the first AMS Book Prize, for |

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### Contents

What Is a Manifold? | 3 |

11 Polygons and Surfaces | 4 |

12 Hyperbolic Surfaces | 7 |

13 The Totality of Surfaces | 17 |

14 Some ThreeManifolds | 31 |

Hyperbolic Geometry and Its Friends | 43 |

21 Negatively Curved Surfaces in Space | 45 |

22 The Inversive Models | 53 |

35 Discrete Groups | 153 |

36 Bundles and Connections | 158 |

37 Contact Structures | 168 |

38 The Eight Model Geometries | 179 |

39 Piecewise Linear Manifolds | 190 |

310 Smoothings | 193 |

The Structure of Discrete Groups | 209 |

42 Euclidean Manifolds and Crystallographic Groups | 221 |

23 The Hyperboloid Model and the Klein Model | 64 |

24 Some Computations in Hyperbolic Space | 74 |

25 Hyperbolic Isometries | 86 |

26 Complex Coordinates for Hyperbolic ThreeSpace | 98 |

27 The Geometry of the ThreeSphere | 103 |

Geometric Manifolds | 109 |

32 Triangulations and Gluings | 118 |

33 Geometric Structures on Manifolds | 125 |

34 The Developing Map and Completeness | 139 |

43 ThreeDimensional Euclidean Manifolds | 231 |

44 Elliptic ThreeManifolds | 242 |

45 The ThickThin Decomposition | 253 |

46 Teichmuller Space | 258 |

47 ThreeManifolds Modeled on Fibered Geometries | 277 |

Glossary | 289 |

295 | |

301 | |