ON THE COMPLEXITY OF THE CLASSIFICATION PROBLEM FOR TORSION-FREE ABELIAN GROUPS OF FINITE RANK

SIMON THOMAS

1. Introduction

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚ^{n} which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at most n can be naturally identified with the set S(ℚ^{n}) of all nontrivial additive subgroups of ℚ^{n}. In 1937, Baer [4] solved the classification problem for the class S(ℚ) of rank 1 groups as follows.

Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ∈ G, then the p-height of x is defined to be

h_{x}(p) = sup{n ∈ ℕ | There exists y ∈ G such that p^{n}y = x } ∈ ℕ ∪ {∞};

and the characteristic χ(x) of x is defined to be the function

⟨h_{x}(p) | p ∈ ℙ⟩ ∈ (ℕ ∪ {∞})^{ℙ}.

Two functions χ_{1}, χ_{2} ∈ (ℕ ∪ {∞})^{ℙ} are said to be similar or to belong to the same type, written χ_{1} ≡ χ_{2}, iff

(a) χ_{1}(p) = χ_{2}(p) for almost all primes p; and

(b) if χ_{1}(p) ≠ χ_{2}(p), then both χ_{1}(p) and χ_{2}(p) are finite. Clearly ≡ is an equivalence relation on (ℕ ∪ {∞})^{ℙ}. If G is a torsion-free abelian group and 0 ≠ x ∈ G, then the type τ(x) of x is defined to be the ≡-equivalence class containing the characteristic χ(x).

Research partially supported by NSF Grants.

2 SIMON THOMAS

Now suppose that G ∈ S(ℚ) is a rank 1 group. Then it is easily checked that τ(x) = τ(y) for all 0 ≠ x, y ∈ G. Hence we can define the type τ(G) of G to be τ(x), where x is any non-zero element of G. In [4], Baer proved that τ(G) is a complete invariant for the isomorphism problem for S(ℚ).

Theorem 1.1 (Baer [4]). If G, H ∈ S(ℚ), then G ≅ H iff τ(G) = τ(H).

However, the situation is much less satisfactory in the case of the torsion-free abelian groups of rank n ≥ 2. In the late 1930s, Kurosh [20] and Malcev [21] found complete invariants for these groups consisting of equivalence classes of infinite sequences ⟨M_{p} | p ∈ ℙ⟩ of matrices, where each M_{p} ∈ GL_{n}(ℚ_{p}). However, as Fuchs [11, Section 93] remarks, the associated equivalence relation is so complicated that the problem of deciding whether two sequences are equivalent is as difficult as that of deciding whether the corresponding groups are isomorphic. It is natural to ask whether the classification problem for S(ℚ^{n}) is genuinely more difficult when n ≥ 2. Of course, if we wish to show that the classification problem for the groups of rank n ≥ 2 is intractible, it is not enough merely to prove that there are 2^{ω} such groups up to isomorphism. For there are 2^{ω} pairwise nonisomorphic groups of rank 1 and we have already pointed out that Baer has given a satisfactory classification for this class of groups. In this paper, following Friedman-Stanley [10] and Hjorth-Kechris [15], we shall use the more sensitive notions of descriptive set theory to measure the complexity of the classification problem for the groups of rank n ≥ 2.

Let X be a standard Borel space; i.e. a Polish space equipped with its associated σ-algebra of Borel subsets. Then a Borel equivalence relation on X is an equivalence relation E ⊆ X^{2} which is a Borel subset of X^{2}. If E, F are Borel equivalence relations on the standard Borel spaces X, Y respectively, then we say that E is Borel reducible to F and write E ≤_{B} F if there exists a Borel function f : X → Y such that xEy iff f(x)F f(y). We say that E and F are Borel bireducible and write E ∼_{B} F if both E ≤_{B} F and F ≤_{B} E. Finally we write E <_{B} F if both E ≤_{B} F and F ≰_{B} E. Most of the Borel equivalence relations that we shall consider in this paper arise from group actions as follows. Let G be a countable group equipped with the discrete topology. Then a standard Borel G-space is a standard Borel space X equipped with a Borel action (g, x) ↦ g.x of G on X. The corresponding G-orbit equivalence relation on X, which we shall denote by E_{G}^{X},

TORSION-FREE ABELIAN GROUPS OF FINITE RANK 3

is a countable Borel equivalence relation; i.e. a Borel equivalence relation E such that every E-equivalence class is countable. Conversely, by Feldman-Moore [9], if E is an arbitrary countable Borel equivalence relation on the standard Borel space X, then there exists a countable group G and a Borel action of G on X such that E = E_{G}^{X}.

To see how the classification problem for torsion-free abelian groups fits into this context, note that S(ℚ^{n}) is a Borel subset of the Polish space P(ℚ^{n}) of all subsets of ℚ^{n} and so S(ℚ^{n}) is a standard Borel space. (Here we are identifying P(ℚ^{n}) with the space 2^{ℚ^{n}} of all functions h : ℚ^{n} → {0, 1} equipped with the product topology.) Furthermore, S(ℚ^{n}) is a standard Borel GL_{n}(ℚ)-space under the action induced from the natural action of GL_{n}(ℚ) on the vector space ℚ^{n}; and it is easily checked that if A, B ∈ S(ℚ^{n}), then A ≅ B iff there exists an element φ ∈ GL_{n}(ℚ) such that φ(A) = B. Thus the isomorphism relation on S(ℚ^{n}) is a countable Borel equivalence relation.

Notation 1.2. ≅_{n} denotes the isomorphism relation on S(ℚ^{n}).

It is clear that (≅_{n}) ≤_{B} (≅_{n+1}) for all n ≥ 1; and our earlier question on the difficulty of the classification problem for the torsion-free abelian groups of rank n ≥ 2 can be rephrased as the question of whether (≅_{1}) <_{B} (≅_{n}) when n ≥ 2. Before we discuss Hjorth’s solution [14] of this problem and some more recent developments in this area, it will be helpful to first give a brief account of some of the general theory of countable Borel equivalence relations. (A detailed development of the theory can be found in Jackson-Kechris-Louveau [16].)

The least complex countable Borel equivalence relations are those which are smooth; i.e. those countable Borel equivalence relations E on a standard Borel space X for which there exists a Borel function f : X → Y into a standard Borel space Y such that xEy iff f(x) = f(y). Next in complexity come those countable Borel equivalence relations E such that E is Borel bireducible with the Vitali equivalence relation E_{0} defined on 2^{ℕ} by xE_{0}y iff x(n) = y(n) for almost all n. More precisely, by Harrington-Kechris-Louveau [13], if E is a countable Borel equivalence relation, then E is non-smooth iff E_{0} ≤_{B} E. Furthermore, by Dougherty-Jackson-Kechris [8], if E is a countable Borel equivalence relation on a standard Borel space X, then the following three properties are equivalent:

4 SIMON THOMAS

(1) E ≤_{B} E_{0}.

(2) E is hyperfinite; i.e. there exists an increasing sequence

F_{0} ⊆ F_{1} ⊆ ⋯ ⊆ F_{n} ⊆ ⋯

of finite Borel equivalence relations on X such that E = ⋃_{n∈ℕ} F_{n}. (Here an equivalence relation F is said to be finite iff every F -equivalence class is finite.)

(3) There exists a Borel action of ℤ on X such that E = E_{ℤ}^{X}.

It turns out that there is also a most complex countable Borel equivalence relation E_{∞}, which is universal in the sense that F ≤_{B} E_{∞} for every countable Borel equivalence relation F, and that E_{0} <_{B} E_{∞}. (Clearly this universality property uniquely determines E_{∞} up to Borel bireducibility.) E_{∞} has a number of natural realisations in many areas of mathematics, including algebra, topology and recursion theory. (See Jackson-Kechris-Louveau [16].) For example, E_{∞} is Borel bireducible to both the isomorphism relation for finitely generated groups [24] and the isomorphism relation for fields of finite transcendence degree [25].

For many years, it was an open problem whether, up to Borel bireducibility, there existed infinitely many countable Borel equivalence relations E such that E_{0} <_{B} E <_{B} E_{∞}. This problem was finally resolved by Adams-Kechris [2], who used Zimmer’s superrigidity theory [27] to show that there are actually 2^{ω} such relations E up to Borel bireducibility. Of these, the only ones which have been extensively studied are the treeable relations, which were originally introduced by Adams [1]. The countable Borel equivalence relation E on the standard Borel space X is said to be treeable iff there exists a Borel relation R ⊆ X × X such that:

(a) ⟨X; R⟩ is an acyclic graph; and

(b) the connected components of ⟨X; R⟩ are precisely the E-equivalence classes. For example, whenever a countable free group F has a free Borel action on a standard Borel space X, then the corresponding orbit equivalence relation E_{F}^{X} is treeable. (Here F is said to act freely on X iff g.x ≠ x for all 1 ≠ g ∈ F and x ∈ X.) The class of treeable relations is not nearly so well understood as that of the hyperfinite relations. For example, it is known that there exists a universal treeable countable Borel equivalence relation E_{T ∞}; but it remains open whether there exists a (necessarily treeable) countable Borel equivalence relation E such

TORSION-FREE ABELIAN GROUPS OF FINITE RANK 5

that

E_{0} <_{B} E <_{B} E_{T ∞},

or even whether E_{T ∞} ≤_{B} E for every non-hyperfinite countable Borel equivalence relation E. (A fuller account of the notion of treeability can be found in Hjorth-Kechris [15] and Jackson-Kechris-Louveau [16].)

Returning to our discussion of the complexity of the isomorphism relation ≅_{n} on S(ℚ^{n}), it is easily checked that Baer’s classification of the rank 1 groups implies that (≅_{1}) ∼_{B} E_{0}. In [15], Hjorth-Kechris conjectured that (≅_{n}) ∼_{B} E_{∞} for all n ≥ 2. Of course, if true, this would have explained the failure to find a satisfactory system of complete invariants for the torsion-free abelian groups of rank n ≥ 2, since nobody expects such a system to exist for the class of finitely generated groups. In [14], Hjorth provided some evidence for this conjecture by proving that if n ≥ 2, then ≅_{n} is not hyperfinite and hence (≅_{1}) <_{B} (≅_{n}). (For n ≥ 3, Hjorth proved the stronger result that ≅_{n} is not treeable. More recently, Kechris [18] has shown that ≅_{2} is also not treeable.) Later in [2], Adams-Kechris used Zimmer’s superrigidity theory [27] to prove the intriguing result that

(≅_{1}^{∗}) <_{B} (≅_{2}^{∗}) <_{B} ⋯ <_{B} (≅_{n}^{∗}) <_{B} ⋯

where (≅_{n}^{∗}) is the restriction of the isomorphism relation to the Borel subset S^{∗}(ℚ^{n}) consisting of the rigid groups A ∈ S(ℚ^{n}). Here an abelian group A is said to be rigid if its only automorphism are the obvious ones: a ↦ a and a ↦ −a. In particular, none of the relations ≅_{n}^{∗} is a universal countable Borel equivalence relation. It was not clear whether the Adams-Kechris result should be regarded as evidence for or against the Hjorth-Kechris conjecture, since very little was known concerning the relationship between ≅_{n}^{∗} and ≅_{m} for n, m ≥ 1. Of course, it is clear that (≅_{n}^{∗}) ≤_{B} (≅_{n}) for all n ≥ 1; and it is easily seen that (≅_{1}^{∗}) ∼_{B} E_{0} and so (≅_{1}^{∗}) ∼_{B} (≅_{1}). But, apart from these easy observations, essentially nothing else was known. Very recently, Thomas [22] disproved the Hjorth-Kechris conjecture by showing that (≅_{3}^{∗}) ≰_{B} (≅_{2}). However, Thomas [22] left open the possibility that there might exist an integer n > 2 such that (≅_{n}) ∼_{B} E_{∞}. This possibility was finally eliminated in Thomas [23].

Theorem 1.3 ([23]). (≅_{n+1}^{∗}) ≰_{B} (≅_{n}) for all n ≥ 1.

6 SIMON THOMAS

As an immediate consequence, we obtain that the classification problem for S(ℚ^{n+1}) is strictly more complex than that for S(ℚ^{n}).

Theorem 1.4 ([23]). (≅_{n}) <_{B} (≅_{n+1}) for all n ≥ 1.

The rest of this paper is organised as follows. In Section 2, we shall sketch the proof of Hjorth’s Theorem. In Section 3, we shall discuss the notion of a cocycle of a group action and sketch the proof of the Adams-Kechris Theorem. Finally in Section 4, we shall sketch the proof of Theorem 1.3. Acknowledgements: I would like to thank David Arnold, Greg Hjorth, Alexander Kechris and Boban Velickovic for very helpful discussions concerning the material in this paper.

2. Invariant probability measures and Hjorth’s Theorem

In this section, we shall sketch the proof of Hjorth’s Theorem that (≅_{1}) <_{B} (≅_{n}) for all n ≥ 2.

Theorem 2.1 (Hjorth [14]). If n ≥ 2, then ≅_{n} is not hyperfinite.

Hjorth’s result ultimately depends upon the fact that ≅_{n} is the orbit equivalence relation of the action of GL_{n}(ℚ) on S(ℚ^{n}) and that GL_{n}(ℚ) is a nonamenable group for all n ≥ 2. (Recall that a countable group G is amenable iff there exists a finitely additive G-invariant probability measure ν : P(G) → [0, 1] defined on every subset of G.) However, these observations alone are not enough to conclude that ≅_{n} is not hyperfinite; since for every countable group G, there exists a standard Borel G-space X such that the relation E_{G}^{X} is smooth. For example, we can take X = G × ℝ equipped with the Borel action g.(h, r) = (gh, r).

Proposition 2.2 ([16]). Let G be a countable nonamenable group and let X be a standard Borel G-space with a invariant probability measure ν. If G acts freely on X, then E_{G}^{X} is not hyperfinite.

Here the probability measure ν on the standard Borel G-space X is said to be invariant iff ν(g(B)) = ν(B) for every g ∈ G and Borel subset B ⊆ X. The probability measure ν on X is said to be nonatomic if ν({x}) = 0 for all x ∈ X. Of course, if the probability measure ν on X is invariant under the free action of an infinite group, then ν is necessarily nonatomic.

TORSION-FREE ABELIAN GROUPS OF FINITE RANK 7

Unfortunately we cannot apply Proposition 2.2 directly to the action of GL_{n}(ℚ) on S(ℚ^{n}). Firstly, the action is not free. In fact, for each A ∈ S(ℚ^{n}), the stabiliser of A in GL_{n}(ℚ) is precisely the automorphism group Aut(A) of A. Secondly, Hjorth has shown that there does not exist a GL_{n}(ℚ)-invariant nonatomic probability measure on S(ℚ^{n}). However, the following result will enable us to work instead with the induced action of the subgroup SL_{n}(ℤ) on a suitably chosen Borel subset of S(ℚ^{n}).

Proposition 2.3 ([16]). Let E be a hyperfinite equivalence relation on the standard Borel space X.

(i) If F is a Borel equivalence relation on X and F ⊆ E, then F is hyperfinite. (ii) If B is a Borel subset of X, then the restriction E ↾ B is hyperfinite.

Fix some n ≥ 2 and let S(ℚ^{n}, ℤ^{n}) be the SL_{n}(ℤ)-invariant Borel subset of S(ℚ^{n}) consisting of those subgroups A such that ℤ^{n} ⩽ A.

Theorem 2.4 ([14]). There exists an SL_{n}(ℤ)-invariant nonatomic probability measure ν on S(ℚ^{n}, ℤ^{n}) such that

ν({A ∈ S(ℚ^{n}, ℤ^{n}) | A is rigid }) = 1.

Before we sketch the proof of Theorem 2.4, we shall show how to derive Hjorth’s Theorem.

Proof of Theorem 2.1. Let S^{∗}(ℚ^{n}, ℤ^{n}) be the Borel subset consisting of the rigid groups A ∈ S(ℚ^{n}, ℤ^{n}); and let F be the orbit equivalence relation of the action of SL_{n}(ℤ) on S^{∗}(ℚ^{n}, ℤ^{n}). Since the stabiliser of each group A ∈ S^{∗}(ℚ^{n}, ℤ^{n}) in SL_{n}(ℤ) is Aut(A) = {±I}, it follows that F is the orbit equivalence relation arising from the induced free action of P SL_{n}(ℤ) = SL_{n}(ℤ)/{±I} on S^{∗}(ℚ^{n}, ℤ^{n}). As P SL_{n}(ℤ) is nonamenable, Proposition 2.2 implies that F is not hyperfinite. Since F ⊆ (≅_{n}↾ S^{∗}(ℚ^{n}, ℤ^{n})), Proposition 2.3 implies that ≅_{n} is also not hyperfinite.

Let S_{n} be the standard Polish space consisting of all subgroups of the quotient group ℚ^{n}/ℤ^{n} and let π : ℚ^{n} → ℚ^{n}/ℤ^{n} be the canonical surjection. Then we can define a Borel bijection S(ℚ^{n}, ℤ^{n}) → S_{n} by A ↦ π(A). Let Γ be the dual group of ℚ^{n}/ℤ^{n}; i.e. the space of all homomorphisms ψ : ℚ^{n}/ℤ^{n} → ℝ/ℤ, equipped with the topology of pointwise convergence and the group operation of pointwise addition.

8 SIMON THOMAS

Since ℚ^{n}/ℤ^{n} is discrete, it follows that Γ is a compact group. Let m be the Haar measure on Γ. Let k : Γ → S_{n} be the Borel map assigning the subgroup

ker(ψ) = {h ∈ ℚ^{n}/ℤ^{n} | ψ(h) = 0}

to each ψ ∈ Γ; and let ν̂ be the probability measure defined on S_{n} by

ν̂(B) = m(k^{−1}(B)) = m({ψ ∈ Γ | ker(ψ) ∈ B})

for each Borel subset B ⊆ S_{n}. Then ν is the corresponding Borel probability measure on S(ℚ^{n}, ℤ^{n}) induced by the Borel bijection A ↦ π(A).

It is clear that ν is nonatomic. Using the uniqueness of the Haar measure m, it follows quite easily that ν is SL_{n}(ℤ)-invariant. (For the details, see Hjorth [14, Lemma 4.2]. Finally the next lemma says that the collection of “obviously rigid” groups has ν-measure 1.

Lemma 2.5 ([14]). Let Z_{n} be the Borel subset consisting of those subgroups A ∈ S(ℚ^{n}, ℤ^{n}) which satisfy the following conditions:

(i) pA ≠ A for all primes p ∈ ℙ. (ii) If a, b ∈ A are linearly independent, then τ(a) ≠ τ(b). Then ν(Z_{n}) = 1.

To see that each A ∈ Z_{n} is rigid, suppose that φ ∈ Aut(A). If 0 ≠ a ∈ A, then τ(φ(a)) = τ(a) and so Condition 2.5(ii) implies that there exists 0 ≠ λ_{a} ∈ ℚ such that φ(a) = λ_{a}.a. It follows easily that there exists a fixed λ ∈ ℚ such that λ_{a} = λ for all 0 ≠ a ∈ A. Since pA ≠ A for all primes p ∈ ℙ, it follows that λ ∈ {±1}.

Now suppose that a ∈ ℚ^{n} ∖ ℤ^{n} and that π(a) has order n > 0 in ℚ^{n}/ℤ^{n}. Let Γ_{a} = {ψ ∈ Γ | ψ(a) = 0}. Then

ν({A ∈ S(ℚ^{n}, ℤ^{n}) | a ∈ A}) = ν̂(Γ_{a}) = [Γ : Γ_{a}] = 1/n.

Furthermore, if a, b ∈ ℚ^{n} ∖ ℤ^{n} are linearly independent, then the events

{A ∈ S(ℚ^{n}, ℤ^{n}) | a ∈ A} and {A ∈ S(ℚ^{n}, ℤ^{n}) | b ∈ A}

are ν-independent.

It follows easily that if p ∈ ℙ, then pA ≠ A for ν-almost all A ∈ S(ℚ^{n}, ℤ^{n}). Now let e_{1}, …, e_{n} be the standard basis of ℚ^{n}; and for each p ∈ ℙ, let

E_{p} = {A ∈ S(ℚ^{n}, ℤ^{n}) | e_{1}/p ∈ A and e_{2}/p ∉ A}.

Then ν(E_{p}) = (p − 1)/p^{2} and the events {E_{p} | p ∈ ℙ} are ν-independent. Since

∑_{p∈ℙ}(p − 1)/p^{2} = ∞,

the Second Borel-Cantelli Lemma [5, Theorem 4.4] implies that

ν( ⋂_{n=1}^{∞} ⋃_{p>n} E_{p}) = 1.

Hence τ(e_{1}) ≠ τ(e_{2}) in ν-almost all A ∈ S(ℚ^{n}, ℤ^{n}). Since SL_{n}(ℤ) acts transitively on the set of ordered pairs of 1-dimensional subspaces of ℚ^{n} and ν is SL_{n}(ℤ)- invariant, it follows easily that

ν({A ∈ S(ℚ^{n}, ℤ^{n}) | τ(a) ≠ τ(b) for all linearly independent a, b ∈ A}) = 1.

This completes the proof of Lemma 2.5.

In the remaining sections of this paper, we shall need to work with ergodic probability measures. The G-invariant probability measure μ on the standard Borel G-space X is said to be ergodic iff for every G-invariant Borel subset B ⊆ X, either μ(B) = 0 or μ(B) = 1. It is well-known that the following two properties are equivalent:

(i) μ is ergodic. (ii) If Y is a standard Borel space and f : X → Y is a G-invariant Borel function,

then there exists a G-invariant Borel subset M ⊆ X with μ(M) = 1 such that f ↾ M is a constant function.

It is easily seen that Hjorth’s measure ν on S(ℚ^{n}, ℤ^{n}) is not ergodic. For example, let p ∈ ℙ be a prime and let

B = {A ∈ S(ℚ^{n}, ℤ^{n}) | ℤ^{n} ⩽ pA}.

Then B is an SL_{n}(ℤ)-invariant Borel subset of S(ℚ^{n}, ℤ^{n}) such that ν(B) = (1/p)^{n}. However, using the ergodic decomposition [26] of the ν-preserving action of SL_{n}(ℤ) on S^{∗}(ℚ^{n}, ℤ^{n}), Adams-Kechris proved the following result.

Theorem 2.6 ([2]). There exists an SL_{n}(ℤ)-invariant Borel subset X_{n} of S(ℚ^{n}, ℤ^{n}) with the following properties.

(i) Each A ∈ X_{n} is rigid.

10 SIMON THOMAS

(ii) There exists an SL_{n}(ℤ)-invariant ergodic nonatomic probability measure μ on X_{n}.

3. Cocycles of group actions and the Adams-Kechris theorem

In this section, we shall sketch the proof of the Adams-Kechris theorem.

Theorem 3.1 (Adams-Kechris [2]). (≅_{n}^{∗}) <_{B} (≅_{n+1}^{∗}) for all n ≥ 1.

Theorem 3.1 is an easy consequence of the following more technical result.

Theorem 3.2 ([2]). Let m ≥ 3 and let X be a standard Borel SL_{m}(ℤ)-space with an invariant ergodic probability measure μ. Suppose that 1 ≤ n < m and that f : X → S^{∗}(ℚ^{n}) is a Borel function such that xE_{SL_{m}(ℤ)}^{X}y implies f(x) ≅_{n}^{∗} f(y). Then there exists an SL_{m}(ℤ)-invariant Borel subset M with μ(M) = 1 such that f maps M into a single ≅_{n}^{∗}-class.

Before we sketch the proof of Theorem 3.2, we shall show how to derive the Adams-Kechris Theorem.

Proof of Theorem 3.1. Theorem 2.1 dealt with the case when n = 1. So suppose that n ≥ 2. By Theorem 2.6, there exists an SL_{n+1}(ℤ)-invariant Borel subset X_{n+1} ⊆ S^{∗}(ℚ^{n+1}) with an invariant ergodic probability measure μ. Suppose that (≅_{n+1}^{∗}) ≤_{B} (≅_{n}^{∗}). Then there exists a Borel function f : X_{n+1} → S^{∗}(ℚ^{n}) such that A ≅_{n+1}^{∗} B iff f(A) ≅_{n}^{∗} f(B). Clearly if A is E_{SL_{n+1}(ℤ)}^{X_{n+1}}-equivalent to B, then A ≅_{n+1}^{∗} B and so f(A) ≅_{n}^{∗} f(B). Hence by Theorem 3.2, there exists an SL_{n+1}(ℤ)- invariant Borel subset M ⊆ X_{n+1} with μ(M) = 1 such that f maps M into a single ≅_{n}^{∗}-class C. But clearly f^{−1}(C) consists of only countably many E_{SL_{n+1}(ℤ)}^{X_{n+1}}-classes, which contradicts the fact that μ is nonatomic. Hence (≅_{n+1}^{∗}) ≰_{B} (≅_{n}^{∗}).

Theorem 3.2 is a consequence of Zimmer’s superrigidity theorems for cocycles[27], which were designed to study orbit equivalence of free measure preserving actions of linear algebraic groups. Roughly speaking, Zimmer [27] used his cocycle superrigidity theorems to show that, under suitable hypotheses, if X is a standard Borel G-space with an invariant ergodic probability measure μ and G acts freely on X, then both the group G and its action on X are “almost completely determined” by the orbit equivalence relation E_{G}^{X} and the measure μ. (Strictly speaking, Zimmer’s results are not quite this strong; but this is an accurate summary of the later

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work of Furman [12].) It was an important insight of Adams-Kechris [2] that Zimmer’s cocycle superrigidity theorems could also be applied in a purely descriptive set-theoretic context.

In order to slightly simplify the various definitions, we shall restrict our discussion of cocycles to the case when both groups are countable. However, we should point out that cocycles are most naturally studied in the wider context of locally compact second countable groups. For example, while the statement of Theorem 3.5 only mentions countable groups, the proof involves algebraic groups over ℝ, ℂ and various finite extensions of the p-adic fields ℚ_{p}. (Clear accounts of the theory of cocycles can be found in Zimmer [27] and Adams-Kechris [2]. In particular, Adams-Kechris [2, Section 2] provides a convenient introduction to the basic techniques and results in this area, written for the non-expert in the ergodic theory of groups.)

Let G be a countable group and let X be a standard Borel G-space with an invariant probability measure μ.

Definition 3.3. If H is a countable group, then a Borel function α : G × X → H is called a cocycle if for all g, h ∈ G and x ∈ X,

α(hg, x) = α(h, g.x)α(g, x).

Cocycles typically arise in the following manner. Suppose that Y is a standard Borel H-space and that H acts freely on Y. If f : X → Y is a Borel function such that xE_{G}^{X}y implies f(x)E_{H}^{Y}f(y), then we can define a Borel cocycle α : G×X → H by letting α(g, x) be the unique element of H such that

α(g, x).f(x) = f(g.x).

Suppose now that B : X → H is a Borel function and that f′ : X → Y is defined by f′(x) = B(x).f(x). Then xE_{G}^{X}y also implies that f′(x)E_{H}^{Y}f′(y); and the corresponding cocycle β : G × X → H satisfies

β(g, x) = B(g.x)α(g, x)B(x)^{−1}

for all g ∈ G and x ∈ X. This observation motivates the following definition.

12 SIMON THOMAS

Definition 3.4. If H is a countable group, then the cocycles α, β : G × X → H are equivalent, written α ∼ β, iff there exist a Borel function B : X → H and a G-invariant Borel subset X_{0} ⊆ X with μ(X_{0}) = 1 such that

β(g, x) = B(g.x)α(g, x)B(x)^{−1}

for all g ∈ G and x ∈ X_{0}.

The proof of Theorem 3.2 is based upon a suitable cocycle reduction result; i.e. a theorem which say that under appropriate hypotheses on G and H, every cocycle α : Γ×X → H is equivalent to a cocycle β such that β(Γ×X) is contained in a finite subgroup of H. To see why such a result might be useful, we shall continue our discussion of the example which was introduced just before Definition 3.4. Suppose now that α ∼ β, where β(G × X) is contained in the finite subgroup K of H. Let B : X → H and X_{0} ⊆ X be as in Definition 3.4; and let f′ : X → Y be the Borel function defined by f′(x) = B(x).f(x). Then for each x ∈ X_{0},

Φ(x) = {β(g, x).f′(x) | g ∈ G}
= {f′(z) | zE_{G}^{X}x}

is a nonempty finite set of Y; and clearly if xEy, then Φ(x) = Φ(y). In other words, Φ is a G-invariant Borel map into the standard Borel space Y^{<ω} of all finite subsets of Y. Hence if μ is an ergodic measure on X, then there exists a G-invariant Borel subset M ⊆ X_{0} with μ(M) = 1 such that Φ ↾ M is a constant function; and this implies that f maps M into a single E_{H}^{Y}-class.

Before we state our main cocycle reduction theorem, we need to review some notions from the theory of algebraic groups. Throughout this paper, let Ω be a fixed algebraically closed field of characteristic 0 which contains ℝ and all of the p-adic fields ℚ_{p}. By an algebraic group, we shall always mean a subgroup G of some general linear group GL_{n}(Ω) which is Zariski closed in GL_{n}(Ω). If G can be defined using polynomials with coefficients in the subfield k of Ω, then we say that G is an algebraic k-group. If G is an algebraic k-group, then G(k) denotes the group of all matrices in G with entries in k.

TORSION-FREE ABELIAN GROUPS OF FINITE RANK 13

Theorem 3.5 ([23]). Let m ≥ 3 and let X be a standard Borel SL_{m}(ℤ)-space with an invariant ergodic probability measure. Suppose that G is an algebraic ℚgroup such that dim G < m^{2} − 1 and that H ⩽ G(ℚ). Then for every Borel cocycle α : SL_{m}(ℤ)×X → H, there exists an equivalent cocycle β such that β(SL_{m}(ℤ)×X) is contained in a finite subgroup of H.

Theorem 3.5 is a straightforward consequence of Zimmer’s superrigidity theorem [27, Theorem 5.2.5] and the ideas of Adams-Kechris [2].

Proof of Theorem 3.2. Let m ≥ 3 and let X be a standard Borel SL_{m}(ℤ)-space with an invariant ergodic probability measure μ. Suppose that 1 ≤ n < m and that f : X → S^{∗}(ℚ^{n}) is a Borel function such that xE_{SL_{m}(ℤ)}^{X}y implies f(x) ≅_{n}^{∗} f(y). Since the stabiliser of each group A ∈ S^{∗}(ℚ^{n}) in GL_{n}(ℚ) is Aut(A) = {±I}, it follows that ≅_{n}^{∗} is the orbit equivalence relation arising from the induced free action of H = GL_{n}(ℚ)/{±I} on S^{∗}(ℚ^{n}). Hence we can define a Borel cocycle α : SL_{m}(ℤ) × X → H by

α(g, x) = the unique φ ∈ H such that φ.f(x) = f(g.x).

By Theorem 3.5, there exists an equivalent cocycle β ∼ α such that β(SL_{m}(ℤ)×X) is contained in a finite subgroup of H. Arguing as in the paragraph after Definition 3.4, it follows that there exists an SL_{m}(ℤ)-invariant Borel subset M with μ(M) = 1 such that f maps M into a single ≅_{n}^{∗}-class.

In the next section, we shall also make use of the following cocycle reduction result, which is a special case of Hjorth-Kechris [15, Theorem 10.5]. (While there is no explicit mention of cocycles in the statement of Theorem 3.6, the proof ultimately relies upon a cocycle reduction theorem of Adams-Spatzier [3].)

Theorem 3.6 ([15]). Let m ≥ 3 and let X be a standard Borel SL_{m}(ℤ)-space with an invariant ergodic probability measure μ. Suppose that Y is a standard Borel space and that F is a hyperfinite equivalence relation on Y. If f : X → Y is a Borel function such that xE_{SL_{m}(ℤ)}^{X}y implies f(x)F f(y), then there exists an SL_{m}(ℤ)-invariant Borel subset M with μ(M) = 1 such that f maps M into a single F -class.

14 SIMON THOMAS

4. Torsion-free groups of finite rank

In this section, we shall sketch the proof that (≅_{n}) <_{B} (≅_{n+1}) for all n ≥ 1. Arguing as in the proof of Theorem 3.1, it is enough to prove the following result. (Instead of working directly with S(ℚ^{n}), we shall find it slightly more convenient to work with the Borel subset R(ℚ^{n}) consisting of those subgroups A ∈ S(ℚ^{n}) which have rank exactly n.)

Theorem 4.1 ([23]). Let m ≥ 3 and let X be a standard Borel SL_{m}(ℤ)-space with an invariant ergodic probability measure μ. Suppose that 1 ≤ n < m and that f : X → R(ℚ^{n}) is a Borel function such that xE_{SL_{m}(ℤ)}^{X}y implies f(x) ≅_{n} f(y). Then there exists an SL_{m}(ℤ)-invariant Borel subset M with μ(M) = 1 such that f maps M into a single ≅_{n}-class.

We shall begin by saying a few words about the strategy of the proof of Theorem 4.1. So suppose that m ≥ 3 and that X is a standard Borel SL_{m}(ℤ)-space with an invariant ergodic probability measure μ. Suppose that 1 ≤ n < m and that f : X → R(ℚ^{n}) is a Borel function such that xE_{SL_{m}(ℤ)}^{X}y implies f(x) ≅_{n} f(y). As we pointed out in Section 2, the action of GL_{n}(ℚ) on R(ℚ^{n}) is far from free. In fact, for each A ∈ R(ℚ^{n}), the stabiliser of A in GL_{n}(ℚ) is precisely the automorphism group Aut(A) of A. Consequently, we are initially unable to define a corresponding cocycle α : SL_{m}(ℤ) × X → GL_{n}(ℚ).

In Thomas [22], we were able to get around this difficulty, in the case when n = 2, by reducing to the situation where there exists a Borel subset X_{0} ⊆ X with μ(X_{0}) = 1 such that Aut(f(x)) is a fixed subgroup L of GL_{2}(ℚ) for all x ∈ X_{0}. This meant that ≅_{2}↾ f(X_{0}) was induced by a free action of the quotient group

H = N_{GL_{2}(ℚ)}(L)/L;

and so we could define a corresponding cocycle α : SL_{m}(ℤ) × X → H. However, this would not have been useful unless there existed a suitable cocycle reduction result for cocycles taking values in H; and for such a result to exist, it was necessary that H should be a “reasonably classical” group. Fortunately, using Król’s analysis [19] of the automorphism groups of the torsion-free abelian groups of rank 2, we were able to explicitly enumerate the possibilities for L; and it turned out that we

TORSION-FREE ABELIAN GROUPS OF FINITE RANK 15

could make a further reduction to the situation when H was a slight variant of P GL_{2}(ℚ).

Now suppose that n > 2. Then it seems likely that we can once again reduce to the situation where there exists a Borel subset X_{0} ⊆ X with μ(X_{0}) = 1 such that Aut(f(x)) is a fixed subgroup L of GL_{n}(ℚ) for all x ∈ X_{0}. Unfortunately, as n gets larger, the possibilities for L become more and more complex. In fact, by Corner’s Theorem [7], L can be isomorphic to the group of units of an arbitrary reduced subring of Mat_{d}(ℚ), where d = ⌊n/2⌋; and it is far from clear whether H = N_{GL_{n}(ℚ)}(L)/L will always be a “reasonably classical” group. (Here a ring is said to be reduced iff its additive subgroup contains no nontrivial divisible subgroups.) In order to avoid these algebraic complications, we shall initially replace the isomorphism relation on R(ℚ^{n}) by the coarser relation of quasi-isomorphism. This relation was first introduced in Jónsson [17], where it was shown that the class of torsion-free abelian groups of finite rank has a better decomposition theory with respect to quasi-isomorphism than with respect to isomorphism. This decomposition theory will not concern us. Rather we shall exploit the fact that much of the number-theoretical complexity of a finite rank torsion-free abelian group is lost when we work with respect to quasi-isomorphism; and this turns out to be enough to ensure that the cocycles that arise in our analysis always take values in a “reasonably classical” group. As a bonus, we obtain that the problem of classifying the torsion-free abelian groups of finite rank up to quasi-isomorphism is also intractible.

Definition 4.2. Suppose that A, B ∈ R(ℚ^{n}). Then A is said to be quasi-contained in B, written A ≺_{n} B, if there exists an integer m > 0 such that mA ⩽ B. If A ≺_{n} B and B ≺_{n} A, then A and B are said to be quasi-equal and we write A ≈_{n} B.

It is well-known that if A ∈ R(ℚ^{n}) and m > 0, then [A : mA] < ∞. This implies that if A, B ∈ R(ℚ^{n}), then A ≈_{n} B iff A ∩ B has finite index in both A and B. It is an easy exercise to show that ≈_{n} is a countable Borel equivalence relation on R(ℚ^{n}). The next result seems to be more difficult. (The proof in [23] makes use of the Kurosh-Malcev [20, 21] p-adic localisation technique.)

Theorem 4.3 ([23]). For each n ≥ 1, the relation ≈_{n} is hyperfinite.

16 SIMON THOMAS

Definition 4.4. Suppose that A, B ∈ R(ℚ^{n}). Then A and B are said to be quasiisomorphic, written A ∼_{n} B, if there exists φ ∈ GL_{n}(ℚ) such that φ(A) ≈_{n} B.

Most of our effort in this section will be devoted to proving the following analogue of Theorem 4.1 for the quasi-isomorphism relation.

Theorem 4.5 ([23]). Let m ≥ 3 and let X be a standard Borel SL_{m}(ℤ)-space with an invariant ergodic probability measure μ. Suppose that 1 ≤ n < m and that f : X → R(ℚ^{n}) is a Borel function such that xE_{SL_{m}(ℤ)}^{X}y implies f(x) ∼_{n} f(y). Then there exists an SL_{m}(ℤ)-invariant Borel subset M with μ(M) = 1 such that f maps M into a single ∼_{n}-class.

Arguing as in the proof of Theorem 3.1, we now easily obtain the folowing result.

Theorem 4.6 ([23]). (∼_{n}) <_{B} (∼_{n+1}) for all n ≥ 1.

Before we sketch the proof of Theorem 4.5, we shall show how to derive Theorem 4.1.

Proof of Theorem 4.1. Suppose that f : X → R(ℚ^{n}) is a Borel function such that xE_{SL_{m}(ℤ)}^{X}y implies f(x) ≅_{n} f(y). Then obviously xE_{SL_{m}(ℤ)}^{X}y implies f(x) ∼_{n} f(y); and so by Theorem 4.5, there exists an SL_{m}(ℤ)-invariant Borel subset Y with μ(Y) = 1 such that f maps Y into a single ∼_{n}-class C. Since C is countable, there exists a Borel subset Z ⊆ Y with μ(Z) > 0 and a fixed group A ∈ C such that f(x) = A for all x ∈ Z. Since μ is ergodic, the SL_{m}(ℤ)-invariant Borel subset M = SL_{m}(ℤ).Z satisfies μ(M) = 1; and clearly f maps M into the ≅_{n}-class containing A.

For each A ∈ R(ℚ^{n}), let [A] be the ≈_{n}-class containing A. During the proof of Theorem 4.5, we shall consider the action of GL_{n}(ℚ) on the set of ≈_{n}-classes. In order to describe the setwise stabiliser in GL_{n}(ℚ) of a ≈_{n}-class [A] , it is necessary to introduce the notions of a quasi-endomorphism and a quasi-automorphism. If A ∈ R(ℚ^{n}), then a linear transformation φ ∈ Mat_{n}(ℚ) is said to be a quasiendomorphism of A iff φ(A) ≺_{n} A. Equivalently, φ is a quasi-endomorphism of A iff there exists an integer m > 0 such that mφ ∈ End(A). It is easily checked that the collection QE(A) of quasi-endomorphisms of A is a ℚ-subalgebra of Mat_{n}(ℚ) and that if A ≈_{n} B, then QE(A) = QE(B). A linear transformation φ ∈ Mat_{n}(ℚ) is said to be a quasi-automorphism of A iff φ is a unit of the ℚ-algebra QE(A). The group of quasi-automorphisms of A is denoted by QAut(A).

Lemma 4.7 ([23]). If A ∈ R(ℚ^{n}), then QAut(A) is the setwise stabiliser of [A] in GL_{n}(ℚ).

Now we are ready to give the proof of Theorem 4.5. So let m ≥ 3 and let X be a standard Borel SL_{m}(ℤ)-space with an invariant ergodic probability measure μ. Suppose that 1 ≤ n < m and that f : X → R(ℚ^{n}) is a Borel function such that xE_{SL_{m}(ℤ)}^{X}y implies f(x) ∼_{n} f(y). Let E = E_{SL_{m}(ℤ)}^{X} and for each x ∈ X, let A_{x} = f(x) ∈ R(ℚ^{n}). First notice that there are only countably possibilities for the ℚ-algebra QE(A_{x}). Hence there exists a Borel subset X_{1} ⊆ X with μ(X_{1}) > 0 and a fixed ℚ-subalgebra S of Mat_{n}(ℚ) such that QE(A_{x}) = S for all x ∈ X_{1}. By the ergodicity of μ, we have that μ(SL_{m}(ℤ).X_{1}) = 1. In order to simplify notation, we shall assume that SL_{m}(ℤ).X_{1} = X. After slightly adjusting f if necessary, we can suppose that QE(A_{x}) = S for all x ∈ X. (More precisely, let c : X → X be a Borel function such that c(x)Ex and c(x) ∈ X_{1} for all x ∈ X. Then we can replace f with f′ = f ∘ c.) In particular, we have that QAut(A_{x}) = S^{∗}, the group of units of S, for each x ∈ X. Now suppose that x, y ∈ X and that xEy. Then A_{x} ∼_{n} A_{y} and so there exists φ ∈ GL_{n}(ℚ) such that φ(A_{x}) ≈_{n} A_{y}. Notice that

φSφ^{−1} = φ QE(A_{x})φ^{−1} = QE(φ(A_{x})) = QE(A_{y}) = S

and so φ ∈ N = N_{GL_{n}(ℚ)}(S). Clearly we also have that φ([A_{x}]) = [A_{y}]; and by Lemma 4.7, for each x ∈ X, the stabiliser of [A_{x}] in GL_{n}(ℚ) is QAut(A_{x}) = S^{∗}. Let H = N/S^{∗} and for each φ ∈ N, let \overline{φ} = φS^{∗}. Then we can define a Borel cocycle α : SL_{m}(ℤ) × X → H by

α(g, x) = the unique element \overline{φ} ∈ H such that φ([A_{x}]) = [A_{g.x}].

Lemma 4.8 ([23]). There exists an algebraic ℚ-group G with dim G < m^{2} −1 such that H ⩽ G(ℚ).

Proof. Recall that Ω denotes a fixed algebraically closed field containing ℝ and all of the p-adic fields ℚ_{p}. Let

Λ = Ω ⊗ S ⊆ Mat_{n}(Ω)

18 SIMON THOMAS

be the associated Ω-algebra. Then Λ is an affine ℚ-variety; and the Cayley-Hamilton Theorem implies that the group of units of Λ is given by

Λ^{∗} = {φ ∈ Λ | det(φ) ≠ 0}.

Thus Λ^{∗} is an algebraic ℚ-group and Λ^{∗}(ℚ) = S^{∗}. Furthermore, by Proposition 1.7 [6], Γ = N_{GL_{n}(Ω)}(Λ) is also an algebraic ℚ-group and clearly Γ(ℚ) = N. By Theorem 6.8 [6], G = Γ/Λ^{∗} is an algebraic ℚ-group and

H = Γ(ℚ)/Λ^{∗}(ℚ) ⩽ G(ℚ).

Finally note that

dim G ≤ dim Γ ≤ dim GL_{n}(Ω) = n^{2} < m^{2} − 1.

By Theorem 3.5, α is equivalent to a cocycle β such that β(SL_{m}(ℤ) × X) is contained in a finite subgroup K of H. To simplify notation, we shall assume that β = α. (Recall from Section 3 that there exists a Borel function B : X → H such that on an SL_{m}(ℤ)-invariant Borel subset X_{0} ⊆ X with μ(X_{0}) = 1, the cocycle β corresponds to the Borel function f′ : X → R(ℚ^{n}) defined by f′(x) = B(x).f(x).) Then for each x ∈ X,

Φ(x) = \{φ([A_{x}]) | \overline{φ} = α(g, x) for some g ∈ G\}
= {[A_{z}] | zEx}

is a nonempty finite set of ≈_{n}-classes; and clearly if xEy, then Φ(x) = Φ(y). By the ergodicity of μ, we can suppose that there exists an integer 1 ≤ k ≤ |K| such that |Φ(x)| = k for all x ∈ X. Now let x ↦ (x_{1}, …, x_{k}) be a Borel function from X to X^{k} such that for each x ∈ X,

(a) x_{i}Ex for all 1 ≤ i ≤ k; and

(b) Φ(x) = {[A_{x_{1}}], …, [A_{x_{k}}]}. Finally let f̃: X → R(ℚ^{n})^{k} be the Borel function defined by

f̃(x) = (A_{x_{1}}, …, A_{x_{k}});

and let F be the countable Borel equivalence relation on R(ℚ)^{k} defined by

(A_{1}, …, A_{k}) F (B_{1}, …, B_{k}) iff {[A_{1}], …, [A_{k}]} = {[B_{1}], …, [B_{k}]}.

Using the fact that ≈_{n} is hyperfinite, it follows easily that F is also hyperfinite. (For example, see Section 1 [16].) Notice that if xEy, then Φ(x) = Φ(y) and so f̃(x)F f̃(y). By Theorem 3.6, there exists an SL_{m}(ℤ)-invariant Borel subset M ⊆ X with μ(M) = 1 such that f̃ maps M into a single F -class; and this implies that f maps M into a single ∼_{n}-class. This completes the proof of Theorem 4.5.

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Mathematics Department, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA

  E-mail address: sthomas@math.rutgers.edu