Anatoly Vershik: Totally non free (TNF) dynamical systems, Invariant random subgroups (IRS) and representation theory

Speaker: Anatoly Vershik

Title:  Totally non free (TNF) dynamical systems, Invariant random subgroups (IRS) and representation theory 

Abstract: A TNF action of a group G roughly speaking is an action of G on the space X such that two different points of the X have different stabilizers in the group. There is a direct reduction of such actions to IRS. In our old papers with S. Kerov we observed that for the infinite symmetric group the values of any indecomposable characters of type II_1 are the measure of the set of fixed points for some action. Later it became clear that those actions are TNF so the problem of the description of characters reduced to the description of TNF actions or IRS for a given group. The main result is the following: there is a transparant correspondence between TNF actions of the group G and irreducible representation of the group $G \times G$ whose restriction on $G \times I$ is factor type II_1. All notions will be explained carefully.