^{1}

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.

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

The concept of superrigidity was introduced by Mostow and Margulis in the context of studying the structure of lattices in Lie groups. Here,

We will leave this first form of rigidity on the back burner and primarily consider instead a second form, initially considered by Zimmer, which is concerned with group

It is natural to ask whether there exists an analog of Zimmer’s theorem in the context of general measure-preserving actions, that is, with the algebraic hypothesis on

This paper is organized as follows. The second section gives some background on Borel cocycles, a key tool in rigidity theory. A slightly weakened version of Ioana’s theorem is stated in the third section. The proof itself is split between Section

In Section

Finally, 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.

We begin by introducing a slightly more expansive notion of orbit equivalence rigidity. If

Following Margulis and Zimmer, we will require the language of Borel cocycles to describe and prove superrigidity results. A

The cocycle condition:

When

In practice, when establishing rigidity, one typically shows that an arbitrary cocycle (arising from a homomorphism of orbits) is equivalent to a trivial cocycle (which therefore arises from a homomorphism of actions). Here, we say that homomorphisms of orbits

The cohomology relation for cocycles:

We close this section by remarking that not all cocycles arise from orbit-preserving maps. An abstract

Cocycle superrigidity results were first established by Margulis and Zimmer for cocycles

Here,

Let

In other words, the conclusion is that

We remark that although our variant of Ioana’s theorem is sufficient for most applications, it is weaker than the state of the art in several ways. First, Ioana requires only that

We begin with the following preliminary result, which roughly speaking says that if

Let

It is easy to see that

Let

We understand this result to say that if

Let

The idea for the conclusion of the proof is as follows. If we had

We actually define

The gap between what he has (the asymptotic information) and what we want (the uniform information) is bridged by Property (T). Once again the first step is to consider an appropriate representation; this time one which compares the values of

More precisely, for each

The remainder of the argument is straightforward. Since the

Now, we simply express

In this section, we use Ioana’s theorem for one of its intended purposes: to find many highly inequivalent actions. The results mentioned here are just meant to give the flavor of applications of superrigidity; they by no means demonstrate the full power of the theorem. In the next section we will discuss the slightly more interesting and difficult application to torsion-free abelian groups. For further applications, see for instance [

In searching for inequivalent actions, one might of course consider a variety of inequivalence notions. Here, we focus on just two of them: orbit inequivalence and Borel incomparability. Recall from the introduction that

Borel bireducibility is a purely set-theoretic notion with its origins in logic. The connection is that if

If

It is elementary to see that neither orbit equivalence or Borel bireducibility implies the other. For instance, given any

We are now ready to begin with the following direct consequence of Ioana’s theorem. It was first established by Simon Thomas in connection working on classification problem for torsion-free abelian groups of finite rank. His proof used Zimmer’s superrigidity theorem and some additional cocycle manipulation techniques; with Ioana’s theorem in hand, the proof will be much simpler.

If

Here,

Let

Now, in the measure-preserving case, it is not difficult to conclude that

Second,

Third, by the ergodicity of

Finally, a short computation confirms the intuitive, algebraic fact that the existence of such a map is ruled out by the mismatch in primes between the left-and right-hand sides. We give just a quick sketch; for a few more details see [

This argument can be easily generalized to give uncountably many incomparable actions of

So far, we have considered only free actions of

Unfortunately, in the purely Borel context it is not sufficient to work with actions which are free almost everywhere, since in this case we are not allowed to just delete a null set on the right-hand side. The next result shows how to get around this difficulty. Once again, it was originally obtained by Simon Thomas using Zimmer’s superrigidity theorem.

If

Here,

First suppose that

The proof in the case of Borel reducibility requires an extra step. Namely, we cannot be sure that

First, let us assume that there exists a conull subset

Note that since

Now, let

If

The idea of the proof is to apply Margulis’s superrigidity theorem. That is, one wishes to conclude that such an embedding lifts to some kind of rational map

The torsion-free abelian groups of rank 1 were classified by Baer in 1937. The next year, Kurosh and Malcev expanded on his methods to give classifications for the torsion-free abelian groups of ranks 2 and higher. Their solution, however, was considered inadequate because the invariants they provided were no easier to distinguish than the groups themselves.

In 1998, Hjorth proved, using methods from the study of Borel equivalence relations, that the classification problem for rank 2 torsion-free abelian groups is

Let

For

Thomas’s original argument used Zimmer’s superrigidity theorem. In this presentation, we have essentially copied his argument verbatim, with a few simplifications stemming from the use of Ioana’s theorem instead of Zimmer’s theorem.

The first connection between this result and the results of the last section is that for

The map

Here, a subgroup of

There exists a Borel reduction from

Since

To verify that it works, one uses the fact that the Kurosh-Malcev construction is

Let

By definition, we have that

One can now formulate a strategy for proving Thomas’s theorem along the following lines:

By passing to a conull subset of

There cannot exist a nontrivial homomorphism from (a finite index subgroup of)

This would yield a contradiction, since by Proposition

Now, rather than attempting to fix the automorphism group of

Thus, we have successfully obtained our analog of Claim

To see this, first note that since the subalgebra