Weak containment in the space of actions of a free group

Abstract

It is shown that the translation action of the free group with n generators on its profinite completion is the maximum, in the sense of weak containment, measure preserving action of this group. Using also a result of Abért–Nikolov this is used to give a new proof of Gaboriau's theorem that the cost of this group is equal to n. A similar maximality result is proved for generalized shift actions. Finally a study is initiated of the class of residually finite, countable groups for which the finite actions are dense in the space of measure preserving actions.

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