Obstructions for quantitative measure equivalence between locally compact groups

TL;DR

The paper extends quantitative measure equivalence from discrete groups to unimodular locally compact groups by introducing $(oldsymbol{unction{ ext{varphi} }},L^0)$-integrable cocycles and analyzing their impact on coarse geometry. It proves that, under a $(oldsymbol{unction{ ext{varphi} }},L^0)$-coupling from $G$ to $H$, the volume growth satisfies $V_G(n) preccurlyeq V_H(oldsymbol{unction{ ext{varphi}^{-1} } }(n))$ and isoperimetric profiles obey $j_{p,H}(n) preccurlyeq j_{p,G}(n)$ (with a $(oldsymbol{unction{ ext{varphi} }},L^0)$-version for $j_{1}$). The work also establishes thresholds showing that certain integrability levels cannot be achieved, i.e., there are no quantitatively critical measure equivalence couplings between appropriate lcsc groups, and provides a preparatory lemma enabling reductions to cross-sections and ergodic arguments. Overall, the results yield rigidity obstructions linking measure equivalence to coarse geometric invariants in the locally compact setting and lay groundwork for further extensions in this framework.

Abstract

Given a measure equivalence coupling between two finitely generated groups, Delabie, Koivisto, Le Maître 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.

Theorems & Definitions (55)

• Definition 1.1
• Definition 1.2: Dye dyeGroupsMeasurePreserving1959
• Definition 1.3
• Theorem 1.4: shalomHarmonicAnalysisCohomology2004
• Theorem 1.5: austinIntegrableMeasureEquivalence2016
• Theorem 1.6: delabieQuantitativeMeasureEquivalence2022
• Theorem 1.7: delabie$mathrmL^p$MeasureEquivalence2025
• Theorem A: see Theorem \ref{['th:volumegrowth']}
• Remark 1.8
• Theorem 1.9: delabieQuantitativeMeasureEquivalence2022