Logic Seminar

Please note that the time is PST

Let GG be a countably infinite group and let SubGSubG be the compact space of subgroups H⩽GH⩽G. Then an invariant random subgroup (IRS) of GG is a probability measure νν on SubGSubG which is invariant under the conjugation action of GG on SubGSubG.

In this talk, after a brief introduction to the theory of invariant random subgroups, I will discuss some of the many basic questions in this relatively new area. For example, if νν is an ergodic IRS of a countable group GG, then we obtain a corresponding zero-one law on SubGSubG for the class of group-theoretic properties ΦΦ such that the set {H∈SubG∣H has property Φ}{H∈SubG∣H has property Φ} is νν-measurable; and thus νν concentrates on a collection of subgroups which are quite difficult to distinguish between. Consequently, it is natural to ask whether there exists an ergodic IRS of a countable group GG which does not concentrate on the subgroups H⩽GH⩽G of a single isomorphism type.