Stability of products of equivalence relations

TL;DR

This paper characterizes when the product of two ergodic p.m.p. equivalence relations is stable, showing it occurs if and only if one component is stable, and discusses related results for McDuff factors.

Contribution

It provides a new local characterization of stable equivalence relations and establishes a criterion for stability of their products, extending understanding in ergodic theory.

Findings

Product stability iff one component is stable

New local characterization of stable relations

Partial results on McDuff II_1 factors

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.