Ergodic Subequivalence Relations Induced by a Bernoulli Action |
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Authors: | Ionut Chifan Adrian Ioana |
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Institution: | (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
G\curvearrowright 0, 1]G{\Gamma\curvearrowright 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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