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Stability of products of equivalence relations
- Part of
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- Journal:
- Compositio Mathematica / Volume 154 / Issue 9 / September 2018
- Published online by Cambridge University Press:
- 17 August 2018, pp. 2005-2019
- Print publication:
- September 2018
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- Article
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An ergodic probability measure preserving (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 
$\text{II}_{1}$ 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 
$\text{II}_{1}$ factors is also discussed and some partial results are given.
