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Relative solidity for biexact groups in measure equivalence

Changying Ding, Daniel Drimbe

TL;DR

The paper proves a relative solidity phenomenon in the measure equivalence (ME) setting for products of biexact groups, yielding a precise UPF-type rigidity: if $\Gamma_1,\dots,\Gamma_n$ are nonamenable biexact and $\times_{k=1}^n\Gamma_k$ is ME to a product $\Lambda_1\times\Lambda_2$, then there is a partition $T_1\sqcup T_2=\{1,\dots,n\}$ with $\Lambda_j\sim_{ME}\times_{k\in T_j}\Gamma_k$. The authors develop an ME-variant of Popa’s intertwining-by-bimodules, study relative amenability via ME couplings, and use comultiplication techniques to transfer intertwinings to ME, together with boundary-piece/biexactness machinery to establish relative solidity in ME. These results yield infinite UPF results, infinite direct-sum rigidity, wreath-product ME rigidity, and ME classifications in broad contexts, including non-weakly amenable biexact groups, thereby extending Ozawa–Popa and Sako-type rigidity phenomena to orbit-equivalence and ME settings. The work broadens the toolkit for measured group theory, enabling robust product-structure detection and decompositions under ME, with implications for fundamental groups of OE relations and wreath products. Overall, the paper strengthens the rigidity landscape in measured group theory by connecting intertwining, solidity, and ME couplings in the biexact regime, and by deriving wide-ranging corollaries for infinite constructions and orbit-equivalence rigidity.

Abstract

We demonstrate a relative solidity property for the product of a nonamenable biexact group with an arbitrary infinite group in the measure equivalence setting. Among other applications, we obtain the following unique product decomposition for products of nonamenable biexact groups, strengthening \cite{Sa09}: for any nonamenable biexact groups $Γ_1,\cdots, Γ_n$, if a product group $Λ_1\times Λ_2$ is measure equivalent to $\times_{k=1}^nΓ_k$, then there exists a partition $T_1\sqcup T_2=\{1,\dots, n\}$ such that $Λ_i$ is measure equivalent to $\times_{k\in T_i}Γ_k$ for $i=1,2$.

Relative solidity for biexact groups in measure equivalence

TL;DR

The paper proves a relative solidity phenomenon in the measure equivalence (ME) setting for products of biexact groups, yielding a precise UPF-type rigidity: if are nonamenable biexact and is ME to a product , then there is a partition with . The authors develop an ME-variant of Popa’s intertwining-by-bimodules, study relative amenability via ME couplings, and use comultiplication techniques to transfer intertwinings to ME, together with boundary-piece/biexactness machinery to establish relative solidity in ME. These results yield infinite UPF results, infinite direct-sum rigidity, wreath-product ME rigidity, and ME classifications in broad contexts, including non-weakly amenable biexact groups, thereby extending Ozawa–Popa and Sako-type rigidity phenomena to orbit-equivalence and ME settings. The work broadens the toolkit for measured group theory, enabling robust product-structure detection and decompositions under ME, with implications for fundamental groups of OE relations and wreath products. Overall, the paper strengthens the rigidity landscape in measured group theory by connecting intertwining, solidity, and ME couplings in the biexact regime, and by deriving wide-ranging corollaries for infinite constructions and orbit-equivalence rigidity.

Abstract

We demonstrate a relative solidity property for the product of a nonamenable biexact group with an arbitrary infinite group in the measure equivalence setting. Among other applications, we obtain the following unique product decomposition for products of nonamenable biexact groups, strengthening \cite{Sa09}: for any nonamenable biexact groups , if a product group is measure equivalent to , then there exists a partition such that is measure equivalent to for .

Paper Structure

This paper contains 16 sections, 26 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Terminology
  4. Intertwining-by-bimodules
  5. Relative amenability
  6. Intertwining and relative amenability via measure equivalence
  7. From intertwining to measure equivalence
  8. Biexact groups and von Neumann algebras
  9. The small-at-infinity boundary and boundary pieces
  10. Biexactness
  11. Relative solidity in measure equivalence
  12. Technical variants
  13. Proofs of Theorem \ref{['theorem.upf.biexact']} and Theorem \ref{['theorem.product.rigidity']}
  14. Proof of Theorem \ref{['theorem.upf.biexact']}
  15. Proof of Theorem \ref{['theorem.product.rigidity']}
  16. ...and 1 more sections

Key Result

Theorem A

Let $\Gamma_1, \dots, \Gamma_n$ be nonamenable biexact groups. Suppose $\times_{k=1}^n\Gamma_k$ is measure equivalent to a product $\Lambda_1\times \Lambda_2$ of i.c.c. groups. Then there exists a partition $T_1\sqcup T_2=\{1,\dots, n\}$ such that $\Lambda_j$ is measure equivalent to $\times_{i\in T

Theorems & Definitions (55)

  • Theorem A
  • Conjecture B
  • Theorem C
  • Theorem D
  • Example 1.1
  • Theorem E
  • Corollary F
  • Theorem 2.1: Po06B
  • Lemma 2.2
  • proof
  • ...and 45 more