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Measure equivalence embeddings of free groups and free group factors

Tey Berendschot, Stefaan Vaes

TL;DR

This work provides an explicit measure equivalence embedding of the free group $\mathbb{F}_2$ into any nonamenable locally compact second countable group $G$, yielding strong ergodic and flow-type conclusions for $G$ and revealing a robust ME framework for II$_1$ factors. The authors connect group ME embeddings to von Neumann algebra ME embeddings via Bernoulli cocycles and random ergodic techniques, showing that nonamenability corresponds to $L(\mathbb{F}_2)$-ME-embeddability and that properties such as (T) and Haagerup are preserved under ME. They further demonstrate that every nonamenable lcsc group admits strongly ergodic actions of any Krieger type and any prescribed flow of weights, illuminating a wide spectrum of dynamical possibilities. The paper also develops a comprehensive operator-algebraic theory of ME embeddings, proves transitivity and inheritance results, and provides examples illustrating the distinction between ME and embeddability, laying groundwork for a deeper understanding of measure-theoretic and von Neumann-algebraic rigidity.

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.

Measure equivalence embeddings of free groups and free group factors

TL;DR

This work provides an explicit measure equivalence embedding of the free group into any nonamenable locally compact second countable group , yielding strong ergodic and flow-type conclusions for and revealing a robust ME framework for II factors. The authors connect group ME embeddings to von Neumann algebra ME embeddings via Bernoulli cocycles and random ergodic techniques, showing that nonamenability corresponds to -ME-embeddability and that properties such as (T) and Haagerup are preserved under ME. They further demonstrate that every nonamenable lcsc group admits strongly ergodic actions of any Krieger type and any prescribed flow of weights, illuminating a wide spectrum of dynamical possibilities. The paper also develops a comprehensive operator-algebraic theory of ME embeddings, proves transitivity and inheritance results, and provides examples illustrating the distinction between ME and embeddability, laying groundwork for a deeper understanding of measure-theoretic and von Neumann-algebraic rigidity.

Abstract

We give a simple and explicit proof that the free group admits a measure equivalence embedding into any nonamenable locally compact second countable (lcsc) group . We use this to prove that every nonamenable lcsc group 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 factors. We prove that a factor is nonamenable if and only if the free group factor admits a measure equivalence embedding into . We prove stability of property (T) and the Haagerup property under measure equivalence of factors.
Paper Structure (17 sections, 27 theorems, 81 equations)

This paper contains 17 sections, 27 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonsingular actions and weak mixing
  4. Correspondences and measure equivalence embeddings
  5. Flow of weights and associated flow
  6. Strong ergodicity
  7. Measure equivalence embeddings of the free groups
  8. Random ergodic theorems
  9. Construction of measure equivalence embeddings
  10. Proof of Theorems \ref{['thm.main-ME-embedding']}, \ref{['thm.ME-embedding-nonamenable']} and \ref{['thm.ME-embedding-noncompact']}
  11. Application: every nonamenable lcsc group admits strongly ergodic actions of any type
  12. Measure equivalence embeddings for II$_1$ factors
  13. Basic results on measure equivalence embeddings
  14. Relation to measure equivalence of groups
  15. Amenability, embeddings of free group factors, Haagerup property
  16. ...and 2 more sections

Key Result

Theorem 1

Let $G$ be an lcsc group. Assume that $\nu$ is a probability measure on $G$ such that $\nu$ is equivalent with the left Haar measure and the convolution operator $\lambda(\nu)$ on $L^2(G)$ has norm less than $1/3$ (the existence of which is equivalent to the nonamenability of $G$). Consider the Bern

Theorems & Definitions (54)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 44 more