## Geometry and Topology Down Under Craig D. Hodgson, William H. Jaco, Martin G. Scharlemann, Stephan Tillmann This book contains the proceedings of the conference Geometry & Topology Down Under, held July 11-22, 2011, at the University of Melbourne, Parkville, Australia, in honour of Hyam Rubinstein. The main topic of the book is low-dimensional geometry and topology. It includes both survey articles based on courses presented at the conferences and research articles devoted to important questions in low-dimensional geometry. Together, these contributions show how methods from different fields of mathematics contribute to the study of 3-manifolds and Gromov hyperbolic groups. It also contains a list of favorite problems by Hyam Rubinstein. |

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### Contents

What is an Almost Normal Surface? | 1 |

The Ergodic Theory of Hyperbolic Groups | 15 |

Mapping Class Groups of 3Manifolds Then and Now | 53 |

Stacks of Hyperbolic Spaces and Ends of 3Manifolds | 65 |

Harmonic Maps and Integrable Systems | 139 |

Some of Hyams Favourite Problems | 165 |

Almost Normal Surfaces with Boundary | 177 |

Algorithms Heuristics and Implementations | 195 |

Seiferters and Branched Coverings | 235 |

Commensurability of Knots and ²Invariants | 263 |

The Groups of Fibred 2Knots | 281 |

On the Number of Hyperbolic 3Manifolds of a Given Volume | 295 |

Seifert Fibered Surgery and Rasmussen Invariant | 321 |

Existence of Spherical Angle Structures on 3Manifolds | 337 |

3Manifolds with Heegaard Splittings of Distance Two | 341 |

Generating the Genus +1 Goeritz Group of a Genus Handlebody | 347 |

### Common terms and phrases

2-sphere algorithm angle structures boundary bounded distance compact component compressing disks compute conjecture connected constant construction corresponding curvature cusp cyclic decomposition define deﬁned Dehn fillings Dehn surgery denote diffeomorphism disjoint edge edge-compressing disk embedded ending lamination endpoints equivalent fibred Figure finite follows function fundamental group genus geodesic Geom geometric given graph Haken handlebody harmonic maps Heegaard splittings hence homeomorphic homotopy Hyam Rubinstein hyperbolic 3-manifolds hyperbolic group integers intersects irreducible isometry isotopy Lemma length linear manifolds Math Mathematics metric minimal Montesinos Montesinos link normal surfaces Note obtained orientable pair parameters path proof Proposition quasi-isometry quasigeodesic R(oo rational tangle Regina resp Seifert fibered Seifert surgeries sequence space spectral curve stack subgroup subset Suppose tangle Teichmüller tetrahedron Theorem topology torus knot triangulation trivial vertex vertices volume ΛΣ