Orbit Structures of ergodic group actions

Let G be a countable group acting ergodically by measure-preserving transformations on a standard probability space (X,m). The orbits of the action define an equivalence relation R on X (understood mod null sets). What information about the group G and its action on X is hidden in the relation R ? In the talk, we shall discuss several recent and older results about lattices in Lie groups, L^2-Betti numbers, property (T) and more. This area of research is also related to other fields such as von Neumann algebras and logic.