A new characterization of amenable groups

Sherman, Jon

doi:10.1090/S0002-9947-1979-0539924-7

A new characterization of amenable groups
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Trans. Amer. Math. Soc. 254 (1979), 365-389 Request permission

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.

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Paradoxical sets and amenability in groups