Table of Contents
ISRN Algebra
Volume 2013, Article ID 387540, 8 pages
Research Article

Ioana's Superrigidity Theorem and Orbit Equivalence Relations

Department of Mathematics, Boise State University, 1910 University DR, Boise, ID 83725, USA

Received 9 October 2013; Accepted 10 November 2013

Academic Editors: M. Przybylska and A. Rapinchuk

Copyright © 2013 Samuel Coskey. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 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.