Bernoulli actions are weakly contained in any free action

Abért, Miklós and Weiss, B. (2013) Bernoulli actions are weakly contained in any free action. ERGODIC THEORY AND DYNAMICAL SYSTEMS, 33 (2). pp. 323-333. ISSN 0143-3857

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Abstract

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. © 2012 Cambridge University Press.

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