Seminar at SMU Delhi

September 16, 2020 (Wednesday) , 3:30 PM at Webinar
Speaker: Romain Tessera, University of Paris
Title: Quantitative measure equivalence
Abstract of Talk
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 Orstein Weiss, all countable amenable groups are measure equivalent, meaning that geometry is completely rubbed out by this equivalence relation. Recently some more restrictive notions have been introduced such as L^p-measure equivalence, where the associated cocycles are assumed to be L^p-integrable. By contrast, a lot of surprising rigidity results have been proved: for instance Bowen has shown that the volume growth is invariant under L^1-measure equivalence, and Austin proved that nilpotent groups that are L^1-measure equivalent have bi-Lipschitz asymptotic cones. In this work we extend this study by trying to understand more systematically how the geometry survives through measure equivalence when an integrability condition is imposed. Our results go in two directions: we prove rigidity results, culminating for amenable groups with a general monotonicity result for the isoperimetric profile, and flexibility results showing that in many instances, the previous result is close to being optimal. We also prove a rigidity result for hyperbolic groups, showing the optimality of a result of Shalom for lattices in SO(n,1).