Abstract

I will talk about measure-preserving actions of countable discrete groups on probability spaces. Classically, one mainly considers two notions of equivalence of such actions, namely conjugacy (or isomorphism) and orbit equivalence, both of which have nice descriptions in the language of von Neumann algebras. I will briefly discuss this classical framework before going into some new notions of equivalence of actions. These are approximate versions of conjugacy and orbit equivalence that were recently introduced and investigated by Sorin Popa and myself, and which can most easily be defined in terms of ultrapowers of von Neumann algebras. I will discuss superrigidity within this new framework, and will also compare approximate conjugacy to (classical) conjugacy for actions of various classes of groups. This talk is based on joint work with Sorin Popa.