  Bader, Furman and Sauer have introduced the notion of integrable measure
equivalence for finitely-generated groups. This 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 classification
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,
albeit in a very weak sense, to bi-Lipschitz maps of asymptotic cones.
