Ioana's superrigidity theorem and orbit equivalence relations

In this expository article, we give a survey of Adrian Ioana’s cocycle superrigidity theorem for profinite actions of Property (T) groups, and its applications to ergodic theory and set theory. In addition to a statement and proof of Ioana’s theorem, this article features:

• An introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems;
• Some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and
• A streamlined proof of Simon Thomas’s theorem that the classification of torsion-free abelian groups of finite rank is intractable.