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Transactions of the American Mathematical Society

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A new characterization of amenable groups

Author: Jon Sherman
Journal: Trans. Amer. Math. Soc. 254 (1979), 365-389
MSC: Primary 43A07; Secondary 28A12, 47D05
MathSciNet review: 539924
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Abstract: Paradoxical sets, which are a natural generalization of the type of sets made famous as Hausdorff-Banach-Tarski paradoxes, are defined in terms of piecewise translations. Piecewise translations are the generalization to arbitrary discrete groups of the maps used in the Banach-Tarski paradoxes as congruences by finite decomposition. A subset of a group is defined to be large if finitely many translates of it can cover the group. The main result of this paper is that a group is amenable if and only if it does not contain a large paradoxical set.

References [Enhancements On Off] (What's this?)

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Keywords: Hausdorff paradox, Banach-Tarski paradox, Banach measure, finitely additive, translation-invariant measure, congruence by finite decomposition, paradoxical set, amenable group, invariant mean, Hahn-Banach extension
Article copyright: © Copyright 1979 American Mathematical Society