Abstract: 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.
Further notes: This article provides a complement to my dissertation. It helps explain some of the context and motivation, and it reproduces a proof of one of the central black boxes that I used.