Organised in collaboration with the OSU Logic seminar
Speaker: Anush Tserunyan (UIUC)
Title: Probability groups as an alternative to Furstenberg’s correspondence
Abstract: Multiple recurrence results for amenable groups are most commonly proven via the Furstenberg correspondence principle, which allows for switching from the hard-to-work-with translation action of \Gamma on (\Gamma, d), where d is an upper density function, to a more friendly measure-preserving action of \Gamma on a (genuine) probability space (X, \mu). I suggest an alternative correspondence principle, where we switch the group itself to a more infinitary group G, which, nevertheless, is equipped with a translation-invariant (genuine) probability measure \mu, and the corresponding action is simply the translation action of G on itself. The object (G, \mu) is an example of, what I call, a probability group, the class of which includes all compact groups and is closed under ultraproducts. One of the advantages of studying measure-preserving actions of probability groups over those of amenable groups is that we can integrate over the group, which makes many statements, such as the mean ergodic theorem, boil down to Fubini’s theorem. As an example, we will give a short proof of a triple recurrence result proved by Bergelson–Tao for quasirandom groups.