Ergodic Subequivalence Relations Induced by a Bernoulli Action

Abstract

Let $\Gamma$ be a countable group and denote by $\Cal S$ the equivalence relation induced by the Bernoulli action $\Gamma \curvearrowright [0,1{]}^{\Gamma }$, where $[0,1{]}^{\Gamma }$ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation $\Cal R$ of $\Cal S$, there exists a partition $\{X_i{\}}_{i\ge 0}$ of $[0,1{]}^{\Gamma }$ with $\Cal R$-invariant measurable sets such that $\Cal R_{|X_0}$ is hyperfinite and $\Cal R_{|X_i}$ is strongly ergodic (hence ergodic), for every $i\ge 1$.

16 pages

References in corpus (1)

• Ends in free minimal spanning forests

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