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.
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The second author was supported by a Clay Research Fellowship.
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Chifan, I., Ioana, A. Ergodic Subequivalence Relations Induced by a Bernoulli Action. Geom. Funct. Anal. 20, 53–67 (2010). https://doi.org/10.1007/s00039-010-0058-7
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DOI: https://doi.org/10.1007/s00039-010-0058-7