Quantitative measure equivalence between amenable groups

TL;DR

This paper develops a quantitative framework for measure and orbit equivalence among amenable groups, revealing rigidity and flexibility phenomena, and introduces new tools like F{46}lner tilings to analyze invariants and embeddings.

Contribution

It establishes invariance and bounds related to the isoperimetric profile under measure equivalence and constructs explicit orbit equivalences with prescribed integrability conditions.

Findings

Isoperimetric profile is invariant under L^1 measure equivalence.

Constructed explicit orbit equivalences with specific integrability properties.

Showed that asymptotic dimension and finite presentability are not preserved under L^1 orbit equivalence.

Abstract

We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of $\mathrm L^p$ measure equivalence. In this paper, our main focus will be on amenable groups, for which we prove both rigidity and flexibility results. On the rigidity side, we prove a general monotonicity property satisfied by the isoperimetric profile, which implies in particular its invariance under $\mathrm L^1$ measure equivalence. This yields explicit "lower bounds" on how integrable a measure coupling between two amenable groups can be. This result also has an unexpected application to geometric group theory: the isoperimetric profile turns out to be monotonous under coarse embedding between amenable groups. This has various applications, among which the existence of an uncountable family of $3$-solvable groups which pairwise do not coarsely embed into one another. On the flexibility side, we construct explicit orbit equivalences between amenable groups with prescribed integrability conditions. Our main tool is a new notion of F{\o}lner tiling sequences. We show in a number of instances that the bounds derived from the isoperimetric profile are sharp up to a logarithmic factor. We also deduce from this study that two important quasi-isometry invariants are not preserved under $\mathrm L^1$ orbit equivalence: the asymptotic dimension and finite presentability.