Ioana's Superrigidity Theorem and Orbit Equivalence Relations

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. 1. Introduction In the past fifteen years superrigidity theory has had a boom in the number and variety of new applications. Moreover, this has been coupled with a significant advancement in techniques and results. In this paper, we survey one such new result, namely, Ioana’s theorem on profinite actions of Property (T) groups and some of its applications in ergodic theory and in set theory. In the concluding section, 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. Although Ioana’s theorem is relatively recent, it will be of interest to readers who are new to rigidity because the proof is natural and there are many immediate applications. Therefore, we were keen to keep the nonexpert in mind. We do assume that the reader is familiar with the notion of ergodicity of a measure-preserving action and with unitary representations of countable groups. We will not go into great detail on Property (T), since for our purposes it is enough to know that satisfies Property (T) when . Rather, we will introduce it just when it is needed, and hopefully its key appearance in the proof of Ioana’s theorem will provide some insight into its meaning. The concept of superrigidity was introduced by Mostow and Margulis in the context of studying the structure of lattices in Lie groups. Here, is said to be a lattice in the (real) Lie group if it is discrete and admits an invariant probability measure. Very roughly speaking, Margulis showed that if is a lattice in a simple (higher-rank) real Lie group , then any homomorphism from into an algebraic group lifts to an algebraic map from to . This implies Mostow’s theorem, which states that any isomorphic lattices in a simple (higher-rank) Lie group must be conjugate