Discrete subgroups of full groups of measure-preserving equivalence relations

It is an interesting open question whether the full group of a hyperfinite equivalence relation (or its not so distant relative, the unitary group of the hyperfinite II_1-factor) can contain a discrete non-abelian free subgroup. Motivated by this, we will discuss various constraints on general countable subgroups in the full group, the nature of the action, and on the measure of fixed point sets, that imply non-discreteness. It turns out that an important condition here is the so-called MIF property (mixed identity-free) which had been studied and used in various other contexts. The talk is based on the recent joint work with Alessandro Carderi, Andreas Thom and Robin Tucker-Drob.