On the absence of quantitatively critical measure equivalence couplings

TL;DR

This paper investigates the limits of integrability for cocycles in measure equivalence couplings between finitely generated groups, showing that certain critical thresholds are not attainable, thus clarifying the boundaries of existing bounds.

Contribution

It provides a negative answer to the open question about the integrability at critical thresholds, demonstrating that these bounds are not sharp in some cases.

Findings

Critical integrability thresholds cannot be achieved in certain group cocycles.

Explicit examples show the non-attainability of the critical $L^p$ bounds.

The results refine understanding of measure equivalence and cocycle integrability limits.

Abstract

Given a measure equivalence coupling between two finitely generated groups, Delabie, Koivisto, Le Ma\^itre and Tessera have found explicit upper bounds on how integrable the associated cocycles can be. These bounds are optimal in many cases but the integrability of the cocycles with respect to these critical thresholds remained unclear. For instance, a cocycle from $\mathbb{Z}^{k+\ell}$ to $\mathbb{Z}^{k}$ can be $\mathrm{L}^p$ for all $p<\frac{k}{k+\ell}$ but not for $p>\frac{k}{k+\ell}$, and the case $p=\frac{k}{k+\ell}$ was an open question which we answer by the negative. Our main result actually yields much more examples where the integrability threshold given by Delabie-Koivisto-Le Ma\^itre-Tessera Theorems cannot be reached.