## Outer Circles: An Introduction to Hyperbolic 3-ManifoldsWe live in a three-dimensional space; what sort of space is it? Can we build it from simple geometric objects? The answers to such questions have been found in the last 30 years, and Outer Circles describes the basic mathematics needed for those answers as well as making clear the grand design of the subject of hyperbolic manifolds as a whole. The purpose of Outer Circles is to provide an account of the contemporary theory, accessible to those with minimal formal background in topology, hyperbolic geometry, and complex analysis. The text explains what is needed, and provides the expertise to use the primary tools to arrive at a thorough understanding of the big picture. This picture is further filled out by numerous exercises and expositions at the ends of the chapters and is complemented by a profusion of high quality illustrations. There is an extensive bibliography for further study. |

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section 5.1 discusses represnetaion varieties and dimension of character varieties.

### Contents

12 | |

Section 2 | 49 |

Section 3 | 76 |

Section 4 | 85 |

Section 5 | 87 |

Section 6 | 105 |

Section 7 | 137 |

Section 8 | 148 |

Section 14 | 263 |

Section 15 | 265 |

Section 16 | 267 |

Section 17 | 268 |

Section 18 | 272 |

Section 19 | 276 |

Section 20 | 279 |

Section 21 | 280 |

Section 9 | 187 |

Section 10 | 232 |

Section 11 | 238 |

Section 12 | 239 |

Section 13 | 250 |

Section 22 | 312 |

Section 23 | 348 |

Section 24 | 364 |

Section 25 | 388 |

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### Common terms and phrases

3-manifold algebraic angle assume automorphism axis boundary component bounded circle closed surface compact compressible cone conformal mapping conjugate converges corresponding curve cusp torus cyclic cylinder deﬁned deﬁnition deformation Dehn surgery Dehn twist denote disk edge elements elliptic embedded endpoints euclidean Exercise ﬁnd ﬁnite number ﬁnite volume ﬁrst ﬁxes freely homotopic fuchsian group fundamental group genus geometric limit geometrically ﬁnite group group G homeomorphic homotopic horoball hyperbolic manifolds hyperbolic metric hyperbolic structure incompressible inﬁnite interior intersection isometric isomorphism kleinian group Lemma length loxodromic manifold M(G measured lamination Möbius transformation mutually disjoint neighborhood nonelementary orbifold orientation preserving orthogonal pair plane polyhedra projection proof punctures quasiconformal mapping quasifuchsian quotient rank two cusp reﬂection Riemann surface satisﬁes Schottky group sequence simple geodesics simple loop simply connected solid cusp space subgroup Suppose Theorem Thurston triangle vertex vertices