Ioana's Superrigidity Theorem and Orbit Equivalence Relations

Samuel Coskey

doi:10.1155/2013/387540

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.

Highlights

In the past fifteen years superrigidity theory has had a boom in the number and variety of new applications
We highlight an application to the classification problem for torsion-free abelian groups of finite rank
The narrative is strictly expository, with most of the material being adapted from the work of Adrian Ioana, mine, and Simon Thomas

Summary

Introduction

In the past fifteen years superrigidity theory has had a boom in the number and variety of new applications. One of the landmark results in this direction was obtained recently by Popa [5], who found a large class of measure-preserving actions Γ ↷ X which are superrigid in the general sense that Γ ↷ X cannot be orbit equivalent to another (free) action without being isomorphic to it. In particular his theorem states that if Γ is a Property (T) group, the free part of its left-shift action on X = 2Γ (the so-called Bernoulli action) is an example of a superrigid action. In the last section, we use Ioana’s theorem to give a self-contained and slightly streamlined proof of Thomas’s theorem that the complexity of the isomorphism problem for torsion-free abelian groups of finite rank increases strictly with the rank

Rigidity via Cocycles

Ioana’s Theorem

Cocycle Untwisting

Ioana’s Proof

Easy Applications

Torsion-Free Abelian Groups of Finite Rank