Stable decompositions and rigidity for products of countable equivalence relations

TL;DR

This paper investigates the stabilization of countable ergodic p.m.p. equivalence relations, revealing conditions for stability and unique decompositions, and introduces new structural insights into product and orbit equivalence relations.

Contribution

It provides the first examples of non-strongly ergodic stable equivalence relations with unique stable decompositions and introduces a new local characterization of the Schmidt property.

Findings

Stabilization of non-Schmidt relations yields stable equivalence relations.

Established a new local characterization of the Schmidt property.

Proved structural results for product and orbit equivalence relations.

Abstract

We show that the stabilization of any countable ergodic p.m.p. equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group, always gives rise to a stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. In the proof, we moreover establish a new local characterization of the Schmidt property. We also prove some new structural results for product equivalence relations and orbit equivalence relations of diagonal product actions.