Inner amenability, property Gamma, McDuff II_1 factors and stable equivalence relations

TL;DR

This paper explores the relationships between inner amenability, property Gamma, McDuff factors, and stable equivalence relations in the context of countable groups and their actions, completing the understanding of their interconnections.

Contribution

It provides the remaining implications and counterexamples to clarify the subtle relations among these properties in operator algebras and ergodic theory.

Findings

Established new implications among properties.

Provided counterexamples to previously conjectured implications.

Completed the classification of relationships among these properties.

Abstract

We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff II_1 factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non implications were obtained in [Ef73,JS85,Va09,Ki12a,Ki12b]. We complete the picture with the remaining implications and counterexamples.