Department of Mathematics

Measure equivalence is an equivalence relation between countable groups that has been introduced by Gromov. A fundamental instance are lattices in a same locally compact group. According to a famous result of Ornstein Weiss, all countable amenable groups are measure equivalent, meaning that geometry is completely rubbed out by this equivalence relation. Recently a more restrictive notion has been investigated called integrable measure equivalence, where the associated cocycles are assumed to be integrable. By contrast, a lot of surprising rigidity results have been proved: for instance Bowen has shown that the volume growth is invariant under integrable measure equivalence, and Austin proved that nilpotent groups that are integrable measure equivalent have bi-Lipschitz asymptotic cones. I will present a work whose goal is to understand more systematically how the geometry survives through measure equivalence when some (possibly very weak) integrability condition is imposed on the cocycles. We shall put the emphasis on amenable groups, for which we will present new rigidity results, and the first flexibility results known in this context. (Contact Person: Dietmar Bisch)