Obstructions for quantitative measure equivalence between locally compact groups

TL;DR

This paper extends bounds on measure equivalence cocycles from finitely generated groups to unimodular locally compact groups, showing the integrability threshold cannot be reached.

Contribution

It generalizes previous results on measure equivalence cocycles to a broader class of locally compact groups and proves the integrability threshold is unattainable.

Findings

Extended bounds to unimodular locally compact groups.

Proved the integrability threshold cannot be achieved.

Generalized previous results on measure equivalence.

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. We extend these results to the broader framework of unimodular compactly generated locally compact groups. We also generalize a result by the first-named author, showing that the integrability threshold described in these statements cannot be achieved.