## 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 | |

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### Common terms and phrases

action acts affine transformations algebra amenable amenable group applied assume Axiom of Choice ball Banach-Tarski Paradox Borel bounded called Chapter choose claim closed compact condition congruences consequence consider consists construction contains contradiction Corollary corresponding countable cube defined Definition denote dense desired discussed disjoint elements equal equidecomposability equivalent example existence exponentially extension fact finitely additive measure fixed points follows four free group free subgroup function G-invariant G-paradoxical given group G growth hence identity implies independent infinite invariant isometries known Lebesgue measure linear locally commutative means metric Moreover nonempty nontrivial normalizes Note obtain paradoxical decomposition partition pieces plane polynomial positive possible preceding presented problem proof Property Proposition proved question relation respect restricted result rotations satisfying semigroup sequence shows similar solvable space sphere square subset Suppose Tarski Theorem theory transformations translation unique unit valid vanishes yields zero