## Banach-Tarski paradox using pieces with the property of Baire.(English)Zbl 0768.04002

The authors prove the following version of the Banach-Tarski paradox which does not depend on the axiom of choice: If $$A$$ and $$B$$ are nonempty bounded open subsets of $$\mathbb{R}^ n$$, $$n\geq 3$$, then there are finitely many pairwise disjoint open subsets $$A_ i$$ of $$A$$ whose union is dense in $$A$$ and isometries $$\rho_ i$$ of $$\mathbb{R}^ n$$ such that $$B_ i=\rho_ i A_ i$$ are pairwise disjoint open subsets of $$B$$ whose union is dense. More generally, if $$X$$ is a separable metric space and $$G$$ is a countable group of homeomorphisms which acts freely on $$X$$ such that some $$\{\rho_ i, \gamma_ i: i=1,2\text{ or } 3\}\subseteq G$$ generate a free subgroup of rank 6, then there are disjoint open sets $$\{R_ i, G_ i: i=1,2,3\}$$ such that $$\bigcup\{\rho_ i R_ i: 1\leq i\leq 3\}$$ and $$\bigcup\{\gamma_ i G_ i: 1\leq i\leq 3\}$$ are dense open subsets of $$X$$. As an application, the authors solve a problem due to E. Marczewski (Problem 1 in S. Wagon: The Banach-Tarski paradox (1985; Zbl 0569.43001)): There is a paradoxical decomposition of the 2-sphere using pieces with the property of Baire. Moreover, the authors announce that the minimal number of 6 such pieces suffice.
Reviewer: N.Brunner (Wien)

### MSC:

 3e+15 Descriptive set theory

Zbl 0569.43001
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