Logic Seminar

A large part of measured group theory studies structural properties of countable groups that hold "on average". This is made precise by studying the orbit equivalence relations induced by free Borel actions of these groups on probability spaces. In this vein, the cyclic (more generally, amenable) groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed quasi-measure-preserving equivalence relation, deriving some of analogues of structural properties of cyclic subgroups of a free group. Most importantly, just like every cyclic subgroup is contained in a unique maximal one, we show that every hyperfinite subequivalence relation is contained in a unique maximal one.