Integrable measure equivalence for groups of polynomial growth

Abstract

Bader, Furman and Sauer have recently introduced the notion of integrable measure equivalence for nitely-generated groups. is is the sub-equivalence relation of measure equivalence obtained by insisting that the relevant cocycles satisfy an integrability condition. They have used it to prove new classication results for hyperbolic groups. The present work shows that groups of polynomial growth are also quite rigid under integrable measure equivalence, in that if two such groups are equivalent then they must have bi-Lipschitz asymptotic cones. This will follow by proving that the cocycles arising from an integrable measure equivalence converge under re-scaling, albeit in a very weak sense, to bi-Lipschitz maps of asymptotic cones.