Operator Theory Seminar
It has been proved by Foreman, Rudolph, and Weiss that classification of measure-preserving transformations up to conjugation is an infeasible task. Long before the descriptive set theoretical viewpoint allowed for estimating the complexity of this classification problem, ergodic theorists shifted their attention to weaker forms of equivalence. The paramount idea here belongs to Dye, who introduced the concept of a full group of an action, which became an important algebraic invariant encoding the orbit partition. The scope of applicability of these ideas grew in a multitude of directions, two of which are most relevant to our talk. First, the concept of a full group was generalized to Borel measure-preserving actions of Polish groups (Carderi and Le Maître) and second, various subgroups of full groups were found to correspond to stronger types of equivalence than the orbit equivalence of actions. The latter phenomena first appeared prominently in Cantor dynamics in the works of Giordano, Putnam, and Skau, who showed that the so-called topological full groups encode flip-conjugacy of minimal homeomorphisms of the Cantor set. Motivated by this correspondence, Le Maître introduced L1 full groups of discrete group actions as the analog of topological full groups within the framework of ergodic theory.
In this talk we will discuss our recent work with Le Maître on the L1 full groups of general Polish group actions. We show that under minor assumptions on the actions, topological derived subgroups of L1 full groups are topologically simple and—when the acting group is locally compact and amenable—are whirly amenable and generically two-generated. For measure-preserving actions of the real line (also known as measure-preserving flows), the topological derived subgroup of an L1 full group coincides with the kernel of the index map, which implies that L1 full groups of free measure-preserving flows are topologically finitely generated if and only if the flow admits finitely many ergodic components. The latter is in a striking contrast to the case of Z-actions, where the number of topological generators is controlled by the entropy of the action.