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Bernoulli actions are weakly contained in any free action

Published online by Cambridge University Press:  07 February 2012

MIKLÓS ABÉRT and
BENJAMIN WEISS
MIKLÓS ABÉRT
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13–15, H-1053, Budapest, Hungary (email: abert@renyi.hu)
BENJAMIN WEISS
Affiliation:
Hebrew University of Jerusalem, Institute of Mathematics, Jerusalem 91904, Israel (email: weiss@math.huji.ac.il)
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Abstract

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Let Γ be a countable group and let f be a free probability measure-preserving action of Γ. We show that all Bernoulli actions of Γ are weakly contained in f. It follows that for a finitely generated group Γ, the cost is maximal on Bernoulli actions for Γ and that all free factors of i.i.d. (independent and identically distributed) actions of Γ have the same cost. We also show that if f is ergodic, but not strongly ergodic, then f is weakly equivalent to f×I, where Idenotes the trivial action of Γ on the unit interval. This leads to a relative version of the Glasner–Weiss dichotomy.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013, pp. 323 - 333
Copyright
©2012 Cambridge University Press

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