Ergodic Subequivalence Relations Induced by a Bernoulli Action |
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Authors: | Ionut Chifan Adrian Ioana |
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Affiliation: | (1) Math.Dept., UCLA, University of California, Los Angeles, CA 90095-155505, USA |
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Abstract: | Let Γ be a countable group and denote by S{mathcal{S}} the equivalence relation induced by the Bernoulli action Gcurvearrowright [0, 1]G{Gammacurvearrowright [0, 1]^{Gamma}}, where [0, 1]Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation R{mathcal{R}} of S{mathcal{S}}, there exists a partition {X i } i≥0 of [0, 1]Γ into R{mathcal{R}}-invariant measurable sets such that R|X0{mathcal{R}_{vert X_{0}}} is hyperfinite and R|Xi{mathcal{R}_{vert X_{i}}} is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1. |
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