Measure equivalence embeddings of free groups and free group factors

TL;DR

This paper establishes measure equivalence embeddings of free groups into nonamenable groups and factors, demonstrating their implications for ergodic actions and properties of $II_1$ factors, including stability of key properties.

Contribution

It provides explicit measure equivalence embeddings for free groups into nonamenable groups and extends the concept to $II_1$ factors, linking nonamenability with measure equivalence.

Findings

Free group $_2$ admits measure equivalence embedding into any nonamenable lcsc group.

Nonamenable lcsc groups have strongly ergodic actions of any Krieger type.

Nonamenable $II_1$ factors admit measure equivalence embeddings from free group factors.

Abstract

We give a simple and explicit proof that the free group $\mathbb{F}_2$ admits a measure equivalence embedding into any nonamenable locally compact second countable (lcsc) group $G$. We use this to prove that every nonamenable lcsc group $G$ admits strongly ergodic actions of any possible Krieger type and admits nonamenable, weakly mixing actions with any prescribed flow of weights. We also introduce concepts of measure equivalence and measure equivalence embeddings for $II_1$ factors. We prove that a $II_1$ factor $M$ is nonamenable if and only if the free group factor $L(\mathbb{F}_2)$ admits a measure equivalence embedding into $M$. We prove stability of property (T) and the Haagerup property under measure equivalence of $II_1$ factors.