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A continuum of non-measure equivalent groups

Adrian Ioana, Robin Tucker-Drob

TL;DR

This work constructs a continuum of pairwise non-measure equivalent ICC groups with property (T), all having zero \\ell^2-Betti numbers and being torsion-free. The authors develop a framework of wreath-like products and cocycle rigidity to control measure equivalence, extend Monod–Shalom-type rigidity to infinite direct sums, and combine these tools to distinguish groups by subsets x ⊆ N. The main contribution is showing that, even in the regime where standard invariants vanish, one can realize uncountably many finitely generated, rigid groups not ME-equivalent to each other, enriching the landscape of measure equivalence rigidity. This provides new avenues for understanding how structural properties such as (T) interact with ME-invariants in the uncountable setting.

Abstract

We construct a continuum sized family $\{G_x\}_{x\in\{0,1\}^{\mathbb N}}$ of pairwise non-measure equivalent countable groups which have property (T) (hence are finitely generated), have zero $\ell^2$-Betti numbers of all orders, and are torsion-free.

A continuum of non-measure equivalent groups

TL;DR

This work constructs a continuum of pairwise non-measure equivalent ICC groups with property (T), all having zero \\ell^2-Betti numbers and being torsion-free. The authors develop a framework of wreath-like products and cocycle rigidity to control measure equivalence, extend Monod–Shalom-type rigidity to infinite direct sums, and combine these tools to distinguish groups by subsets x ⊆ N. The main contribution is showing that, even in the regime where standard invariants vanish, one can realize uncountably many finitely generated, rigid groups not ME-equivalent to each other, enriching the landscape of measure equivalence rigidity. This provides new avenues for understanding how structural properties such as (T) interact with ME-invariants in the uncountable setting.

Abstract

We construct a continuum sized family of pairwise non-measure equivalent countable groups which have property (T) (hence are finitely generated), have zero -Betti numbers of all orders, and are torsion-free.

Paper Structure

This paper contains 10 sections, 20 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Terminology
  4. Measure equivalence
  5. Essentially trivial, amenable and elementary cocycles
  6. Wreath-like product groups
  7. Cocycle rigidity
  8. An extension of results of Monod-Shalom
  9. Pairwise non-ME groups in class $\mathcal{C}$
  10. Proof of Theorem \ref{['A']}

Key Result

Theorem 1.1

There exists a continuum sized family $\{G_x\}_{x\in \{0,1\}^\mathbb N}$ of pairwise non-measure equivalent ICC countable groups which have property (T) (and hence are finitely generated). Moreover, each $G_x$ satisfies $\beta^{(2)}_k(G_x)=0$, for every $k\geq 1$, and can be taken torsion-free.

Theorems & Definitions (43)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • Corollary 3.2
  • Lemma 3.3
  • ...and 33 more