CA : Equivalence relations and measured group theory

Given a measure-preserving action of a group on a finite measure space X, one can consider the equivalence relation of being in the same orbit on X. A central theme in measured group theory is to understand algebraic properties of groups solely through the orbit equivalence relations arising from their actions. At this level, some groups (such as abelian groups) become indistinguishable, while for others the opposite happens: the group and the action can be completely recovered.

The purpose of the course is to become familiar with measured equivalence relations and their interactions with diverse areas of mathematics, including operator algebras, ergodic theory, and descriptive set theory. The field has seen spectacular progress over the last thirty years and continues to generate a great deal of active research. Another aim of the course is to introduce some of the contemporary questions that are currently driving developments in the area.

Bibliographie :
– Alex Furman. Orbit equivalence rigidity. Ann. of Math. (2), 150(3):1083–1108, 1999.
– Donald S. Ornstein and Benjamin Weiss. Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.), 2(1):161–164, 1980.