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arXiv · 1709.00357

Stability of products of equivalence relations

Amine Marrakchi
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Abstract

An ergodic p.m.p. equivalence relation $ \mathcal{R}$ is said to be stable if $\mathcal{R} \cong \mathcal{R} \times \mathcal{R}_0$ where $\mathcal{R}_0$ is the unique hyperfinite ergodic type $\mathrm{II}_1$ equivalence relation. We prove that a direct product $\mathcal{R} \times \mathcal{S}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components $\mathcal{R}$ or $\mathcal{S}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff $\mathrm{II}_1$ factors is also discussed and some partial results are given.

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Amine Marrakchi. 2017-09-01. Stability of products of equivalence relations. https://doi.org/10.1112/s0010437x18007388

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