Ioana's superrigidity theorem and orbit equivalence relations

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.

This article is expository

References in corpus (4)

• Cocycle superrigidity for profinite actions of irreducible lattices
• Orbit equivalence rigidity for ergodic actions of the mapping class group
• The classification of torsion-free abelian groups of finite rank up to isomorphism and up to quasi-isomorphism
• Borel reductions of profinite actions of SL(n,Z)