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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0539924-7
PII: S 0002-9947(1979)0539924-7
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