, Volume 20, Issue 1, pp 53-67
Date: 09 Apr 2010

Ergodic Subequivalence Relations Induced by a Bernoulli Action

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Abstract

Let Γ be a countable group and denote by \({\mathcal{S}}\) the equivalence relation induced by the Bernoulli action \({\Gamma\curvearrowright [0, 1]^{\Gamma}}\) , where [0, 1]Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation \({\mathcal{R}}\) of \({\mathcal{S}}\) , there exists a partition {X i } i≥0 of [0, 1]Γ into \({\mathcal{R}}\) -invariant measurable sets such that \({\mathcal{R}_{\vert X_{0}}}\) is hyperfinite and \({\mathcal{R}_{\vert X_{i}}}\) is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.

The second author was supported by a Clay Research Fellowship.