## The Banach-Tarski ParadoxThis volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, and logic. It unifies the results of contemporary research on the paradox and presents several new results including some unusual paradoxes in hyperbolic space. It also provides up to date proofs and discusses many unsolved problems. |

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This is one of the most enjoyable mathematics books I have seen. The collection of results is both unified and fascinating, and the bibliographic information very helpful.

### Contents

Introduction | 3 |

The Hausdorff Paradox | 15 |

The BanachTarski Paradox Duplicating Spheres and Balls | 21 |

Locally Commutative Actions Minimizing the Number of Pieces in a Paradoxical Decomposition | 34 |

Higher Dimensions and NonEuclidean Spaces | 52 |

Free Groups of Large Rank Getting a Continuum of Spheres from One | 73 |

Paradoxes in Low Dimensions | 96 |

The Semigroup of Equidecomposability Types | 109 |

Applications of Amenability Marczewski Measures and Exotic Measures | 165 |

Growth Conditions in Groups and Supramenability | 188 |

The Role of the Axiom of Choice | 207 |

Euclidean Transformation Groups | 222 |

Jordan Measure | 227 |

Unsolved Problems | 229 |

Addendum to Second Printing | 234 |

236 | |

### Common terms and phrases

action amenable group Axiom of Choice Banach Banach-Tarski Paradox bijection Boolean algebra Borel sets cancellation law cardinal Chapter compact Con(ZF construction contains Corollary countable equidecomposability countably additive defined denote dense elements equivalent Euclidean existence exotic measure exponentially bounded Extension Theorem fact finitely additive finitely additive measure follows free group free non-Abelian free subgroup free subsemigroup function G acts G-invariant measure G-paradoxical group G group of isometries group of rank Hausdorff Hausdorff Paradox hence hyperbolic implies independent infinite integer invariant isometry group isometry-invariant Jordan measurable Lebesgue measure left-invariant linear locally commutative m-divisible measurable sets measure zero Mycielski nontrivial fixed points open set pairwise disjoint paradoxical decomposition partition plane polygons problem Property of Baire Proposition proved question regular-open respect result rotations satisfying semigroup sequence shows solvable sphere subgroup of rank Suppose supramenable systems of congruences Tarski Tarski's Theorem topological total measure transformations translation unit cube vanishes yields