Quantitative measure equivalence between amenable groups

Abstract

We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of 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 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

Entropy, virtual Abelianness and Shannon orbit equivalence

“…We then have If one is permitted to use two different acting groups, then one can show much more easily that certain actions that are known not to be integrably orbit equivalent are, in fact, Shannon orbit equivalent. This phenomenon was observed in [6] in the context of measure equivalence for groups. Consider, for example, the odometer Z-action on {0, 1, 2, 3} N and the action of Z 2 on {0, 1} N × {0, 1} N = ({0, 1} × {0, 1}) N implemented on the canonical generators by T × id and id × T , where T denotes the odometer transformation of {0, 1} N .…”