University of Chicago
Department of Mathematics

When a group acts on a space, it defines an equivalence relation "to be in the same orbit". Assume a finite measure is preserved. Forget the action and the group to retain only the equivalence relation. One then asks: "What does it remember?", "Can one recognize which group produced it?", or even better: "Can one find again the action?". I will explain some striking recent and less recent results. We'll be concerned with discrete groups, ergodic theory and geometric invariants like L²Betti numbers.