Ioana's Superrigidity Theorem and Orbit Equivalence Relations. (30th December 2013)

• Record Type: Journal Article
• Authors: Coskey, Samuel
• Other Names: Przybylska M. Academic Editor.; Rapinchuk A. Academic Editor.
• Is Part Of: ISRN algebra. Volume 2013(2013)
• Publication Date: 2013-12-30
• Subjects: Algebra -- Periodicals; Algebra; Periodicals; Electronic journals; 512
• DOI: 10.1155/2013/387540
• Languages: English
• ISSNs: 2090-6285

Abstract

Abstract : 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.