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Weak containment in the space of actions of a free group

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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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  1. Department of Mathematics, California Institute of Technology, Pasadena, CA, 91125, USA

    Alexander S. Kechris

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  1. Alexander S. Kechris
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Correspondence to Alexander S. Kechris.

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Kechris, A.S. Weak containment in the space of actions of a free group. Isr. J. Math. 189, 461–507 (2012). https://doi.org/10.1007/s11856-011-0182-6

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  • DOI: https://doi.org/10.1007/s11856-011-0182-6

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