We give a survey of Adrian Ioana’s cocycle superrigidity theoremfor profinite actions of Property (T) groups and its applications to ergodic theory and set theory in this expository paper. In addition to a statement and proof of Ioana’s theorem, this paper features the following: (i) an introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems; (ii) some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and (iii) a streamlined proof of Simon Thomas’s theorem that the classification of torsion-free abelian groups of finite rank is intractable.
This document was originally published by Hindawi Publishing Corp. in ISRN Algebra. This work is provided under a Creative Commons Attribution 3.0 License. Details regarding the use of this work can be found at: http://creativecommons.org/licenses/by/3.0/. DOI: 10.1155/2013/387540
Coskey, Samuel. (2013). "Ioana's Superrigidity Theorem and Orbit Equivalence Relations". ISRN Algebra, 1-8.