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

SIMON THOMAS

Abstract. We prove that the classification problem for torsion-free abelian groups of rank three is not Borel reducible to that for torsion-free abelian groups of rank two.

1. Introduction

This paper is a contribution to the project [9, 8, 1, 13] 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 1 ≤ r ≤ n can be naturally identified with the set S(ℚ^{n}) of all nontrivial additive subgroups of ℚ^{n}. In 1937, Baer [3] solved the classification problem for the class S(ℚ) of rank one 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.

Research partially supported by NSF Grants.

2 SIMON THOMAS

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).

Now suppose that G ∈ S(ℚ) is a rank one 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 [3], Baer proved that τ(G) is a complete invariant for the isomorphism problem for S(ℚ).

Theorem 1.1 (Baer [3]). 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 [15] and Malcev [16] 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 [7, 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; or whether, on the contrary, there exists an “explicit” map f : S(ℚ^{n}) → S(ℚ) which reduces the classification problem for S(ℚ^{n}) to that for S(ℚ); i.e. which has the property that A ≅ B iff f(A) ≅ f(B). To give a precise formulation of this question, we need to make use of the notion of Borel reducibility.

Let X be a standard Borel space; i.e. a Polish space equipped with its Borel structure. 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 locally compact second countable group. 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}, is a Borel equivalence relation. In fact, by

TORSION-FREE ABELIAN GROUPS OF RANK TWO 3

Kechris [11], E_{G}^{X} is Borel bireducible with a countable Borel equivalence relation; i.e. a Borel equivalence relation E such that every E-equivalence class is countable. Conversely, by Feldman-Moore [6], 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 complexity of the classification problem for S(ℚ^{n}) can be rephrased as the question of whether (≅_{1}) <_{B} (≅_{n}) when n ≥ 2. In order to explain the solution of this problem and to be able to formulate the main open problems in this area, it is first necessary to 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 [10].)

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. By Dougherty-Jackson-Kechris [5], if E is a countable Borel equivalence relation on a standard Borel space X, then the following three properties are equivalent:

(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 is easily checked that the similarity relation ≡ on the space (ℕ∪{∞})^{ℙ} of characteristics is Borel bireducible with E_{0}. Thus we obtain the following characterisation of the complexity of the isomorphism problem for S(ℚ).

Theorem 1.3 (Folklore). (≅_{1}) ∼_{B} E_{0}.

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 [10].) For example, E_{∞} is Borel bireducible to both the isomorphism relation for finitely generated groups [21] and the isomorphism relation for fields of finite transcendence degree [22].

In [9], Hjorth-Kechris conjectured that (≅_{n}) ∼_{B} E_{∞} for all n ≥ 2. Of course, if true, this would explain 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 [8], Hjorth provided some evidence for this conjecture by proving that E_{0} <_{B} (≅_{n}) for all n ≥ 2. (For n ≥ 3, Hjorth proved the stronger result that ≅_{n} is not treeable. More recently, Kechris [13] has shown that ≅_{2} is also not treeable. See [9] or [10] for a discussion of the notion of treeability.) Later in [1], Adams-Kechris used Zimmer’s superrigidity theorem for cocycles [24, Theorem 5.2.5] to prove the intriguing result that

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

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where (≅_{n}^{∗}) is the restriction of the isomorphism relation to the class of rigid torsionfree abelian 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 or not the Adams-Kechris result provided further evidence for 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 using Theorem 1.1, 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. The main result of this paper says that (≅_{3}^{∗}) ≰_{B} (≅_{2}). Thus ≅_{2} is not a universal countable Borel equivalence relation and so the Hjorth-Kechris conjecture is false.

Theorem 1.4. (≅_{3}^{∗}) ≰_{B} (≅_{2}).

As an immediate consequence, we obtain that the classification problem for S(ℚ^{3}) is strictly more complex than that for S(ℚ^{2}).

Corollary 1.5. (≅_{2}) <_{B} (≅_{3}).

Theorem 1.4 is an easy consequence of Theorem 1.6. But before we can state Theorem 1.6, we need to recall some notions from ergodic theory and group theory. Let G be a locally compact second countable group and let X be a standard Borel G-space. Throughout this paper, a probability measure on X will always mean a Borel probability measure; i.e. a measure which is defined on the collection of Borel subsets of X. The probability measure μ on X is said to be nonatomic if μ({x}) = 0 for all x ∈ X; and μ is said to be G-invariant iff μ(g[A]) = μ(A) for every g ∈ G and Borel subset A ⊆ X. The G-invariant probability measure μ is ergodic iff for every G-invariant Borel subset A ⊆ X, either μ(A) = 0 or μ(A) = 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.

Let G be a locally compact second countable group and let π : G → U(H) be a unitary representation of G on the separable Hilbert space H. Then π almost

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admits invariant vectors if for every ε > 0 and every compact subset K ⊆ G, there exists a unit vector v ∈ H such that ||π(g).v − v|| < ε for all g ∈ K. We say that G is a Kazhdan group if for every unitary representation π of G, if π almost admits invariant vectors, then π has a non-zero invariant vector. If G is a connected semisimple ℝ-group, each of whose almost ℝ-simple factors has ℝ-rank at least 2, and Γ is a lattice in G, then Γ is a Kazhdan group. (For example, see Margulis [17] or Zimmer [24].) In particular, SL_{3}(ℤ) is a Kazhdan group.

For later use, recall that a locally compact second countable group G is amenable if there exists a finitely additive G-invariant probablity measure ν : P(G) → [0, 1] defined on every subset of G. During the proof of Theorem 1.6, we shall make use of the fact that if G is either soluble or abelian-by-finite, then G is amenable. (For example, see Wagon [23, Theorem 10.4].)

In the first three sections of this paper, we shall only discuss countable groups equipped with the discrete topology. In Section 4, we shall also need to consider various linear algebraic groups G(K) ⩽ GL_{n}(K), where K is either ℝ or a finite extension of the field ℚ_{p} of p-adic numbers for some prime p. In this case, G(K) is a locally compact second countable group with respect to the Hausdorff topology; i.e. the topology obtained by restricting the natural topology on K^{n^{2}} to G(K). Any topological notions concerning the group G(K) will always refer to the Hausdorff topology.

Theorem 1.6. Let Γ be a countable Kazhdan group and let X be a standard Borel Γ-space with an invariant ergodic probability measure μ. If f : X → S(ℚ^{2}) is a Borel function such that xE_{Γ}^{X}y implies f(x) ≅_{2} f(y), then there exists a Γ-invariant Borel subset M with μ(M) = 1 such that f maps M into a single ≅_{2}-class.

Proof of Theorem 1.4. Let S(ℚ^{3}, ℤ^{3}) be the Borel set consisting of those G ∈ S(ℚ^{3}) such that ℤ^{3} ⩽ G. Then S(ℚ^{3}, ℤ^{3}) is invariant under the action of the subgroup SL_{3}(ℤ) of GL_{3}(ℚ); and building upon earlier work of Hjorth [8], Adams-Kechris [1, Section 6] have shown that there exists an SL_{3}(ℤ)-invariant Borel subset X of S(ℚ^{3}, ℤ^{3}) with the following properties.

(i) Each G ∈ X is rigid. (ii) There exists an SL_{3}(ℤ)-invariant ergodic nonatomic probability measure μ

on X.

TORSION-FREE ABELIAN GROUPS OF RANK TWO 7

Suppose that (≅_{3}^{∗}) ≤_{B} (≅_{2}). Then there exists a Borel function f : X → S(ℚ^{2}) such that G ≅_{3}^{∗} H iff g(G) ≅_{2} g(H). If G is E_{SL_{3}(ℤ)}^{X}-equivalent to H, then G ≅_{3}^{∗} H and so f(G) ≅_{2} f(H). Since SL_{3}(ℤ) is a Kazhdan group, Theorem 1.6 implies that there exists an SL_{3}(ℤ)-invariant Borel subset M with μ(M) = 1 such that f maps M into a single ≅_{2}-class C. But clearly f^{−1}[C] consists of only countably many E_{SL_{3}(ℤ)}^{X}-classes, which contradicts the fact that μ is nonatomic. Hence (≅_{3}^{∗}) ≰_{B} (≅_{2}).

This paper is organised as follows. In Section 2, we shall discuss the notion of a cocycle of a group action and state the two cocycle reduction results which are needed in the proof of Theorem 1.6. In Section 3, we shall proof Theorem 1.6; and in Section 4, we shall prove our main cocycle reduction result.

Finally we shall address the question of whether it is likely that (≅_{n}) <_{B} (≅_{n+1}) for all n ≥ 1. Hjorth’s result that (≅_{1}) <_{B} (≅_{2}) depends essentially upon the fact that SL_{2}(ℤ) is nonamenable, while GL_{1}(ℚ) = ℚ^{∗} is amenable. In this paper, the proof that (≅_{2}) <_{B} (≅_{3}) is based upon the fact that SL_{3}(ℤ) is a Kazhdan group, while GL_{2}(ℚ) does not contain any infinite Kazhdan subgroups. However, we could also have based our proof upon Zimmer’s superrigidity theorem for cocycles [24, Theorem 5.2.5], which can be used to distinguish between SL_{n}(ℤ)-spaces and SL_{n+1}(ℤ)-spaces for all n ≥ 2. In fact, the main obstruction to proving that (≅_{n}) <_{B} (≅_{n+1}) for n ≥ 3 lies in the field of abelian group theory. The proof of Theorem 1.6 makes heavy use of Król’s analysis [14] of the automorphism groups and endomorphism rings of the torsion-free abelian groups of rank two; and no such analysis has yet been made for the groups of rank n ≥ 3.

Conjecture 1.7. (≅_{n}) <_{B} (≅_{n+1}) for all n ≥ 1.

2. Cocycles

In this section, we shall discuss the notion of a cocycle of a group action and state the two cocycle reduction results which are needed in the proof of Theorem 1.6. (Clear accounts of the theory of cocycles can be found in Zimmer [24] and Adams-Kechris [1]. In particular, Adams-Kechris [1, Section 2] provides a convenient introduction to the basic techniques and results in this area, written for the

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non-expert in the ergodic theory of groups.) Let Γ be a countable group and let X be a standard Borel Γ-space with an invariant probability measure μ.

Definition 2.1. If H is a locally compact second countable group, then a Borel function α : Γ × X → H is called a cocycle if for all g, h ∈ Γ and x ∈ X,

α(hg, x) = α(h, g.x)α(g, x).

(Since we shall only be considering the case when Γ is a countable group, we can work with strict cocycles throughout this paper. For example, see the discussion on page 67 of Zimmer [24].)

In this paper, cocycles will usually arise in the following fashion. Suppose that H is also a countable group and that Y is a standard Borel H-space. Let f : X → Y be a Borel function such that xE_{Γ}^{X}y implies f(x)E_{H}^{Y}f(y). If H acts freely on Y, then we can define a Borel cocycle α : Γ × 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_{Γ}^{X}y also implies f′E_{H}^{Y}f′(y); and the corresponding cocycle α′ : Γ × X → H satisfies

α′(g, x) = B(g.x)α(g, x)B(x)^{−1}

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

Definition 2.2. Let H be a locally compact second countable group. Then the cocycles α, β : Γ × X → H are equivalent, written α ∼ β, iff there exists a Borel function B : X → H such that for all g ∈ Γ,

β(g, x) = B(g.x)α(g, x)B(x)^{−1} μ-a.e.

A cocycle reduction result says that under suitable hypotheses, every cocycle α : Γ × X → H is equivalent to a cocycle β such that β[Γ × X] is contained in a “small” subgroup of H. To see why this might be useful, we shall continue our discussion of the example which was introduced just before Definition 2.2. Suppose now that β[Γ×X] is the identity subgroup of H. Then the associated Borel function f′ : X → Y is Γ-invariant. Hence if μ is an ergodic measure on X, then there exists

TORSION-FREE ABELIAN GROUPS OF RANK TWO 9

a Γ-invariant Borel subset M with μ(M) = 1 such that f′ ↾ M is a constant function; and this implies that f maps M into a single E_{H}^{Y}-class. With a little more effort, we can reach the same conclusion if we only assume that β[Γ × X] is contained in a finite subgroup of H.

The proof of Theorem 1.6 will be based on the following cocycle reduction result, which we shall prove in Section 4.

Theorem 2.3. Let Γ be a countable Kazhdan group and let X be a standard Borel Γ-space with an invariant ergodic probablity measure μ. Then for every Borel cocycle α : Γ × X → P GL_{2}(ℚ), there exists an equivalent cocycle γ such that γ[Γ × X] is contained in a finite subgroup of P GL_{2}(ℚ).

At this point, we can explain the strategy of the proof of Theorem 1.6. So suppose that Γ is a countable Kazhdan group and that X is a standard Borel Γspace with an invariant ergodic probablity measure μ. Let f : X → S(ℚ^{2}) be a Borel function such that xE_{Γ}^{X}y implies f(x) ≅_{2} f(y). Because GL_{2}(ℚ) does not act freely on S(ℚ^{2}), initially we are unable to define a corresponding cocycle α : Γ×X → GL_{2}(ℚ). Instead we attempt to find a Borel subset X_{0} with μ(X_{0}) = 1 such that Aut(f(x)) is a fixed subgroup L of GL_{2}(ℚ) for all x ∈ X_{0}. If we succeed, then ≅_{2}↾ f[X_{0}] is induced by a free action of the quotient group

H = N_{GL_{2}(ℚ)}(L)/L;

and we can define a corresponding cocycle α : Γ × X → H. Then this case can be dealt with using Theorem 2.3. (At first glance, it might appear that we require a whole series of cocycle reduction results, one for each of the possible groups L. But fortunately we can get by with Theorem 2.3.) On the other hand, in those cases when we fail, it turns out that there exists a Borel subset X_{0} with μ(X_{0}) = 1 such that ≅_{2}↾ f[X_{0}] is induced by an action of a countable amenable group H. Then the following result deals with this case.

Theorem 2.4. Let Γ be a countable Kazhdan group and let X be a standard Borel Γ-space with an invariant ergodic probablity measure μ. Suppose that H is a countable amenable group and that Y is a standard Borel H-space. If f : X → Y is a Borel function such that xE_{Γ}^{X}y implies f(x)E_{H}^{Y}f(y), then there exists a Γ-invariant Borel subset M with μ(M) = 1 such that f maps M into a single E_{H}^{Y}-class.

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Proof. By Connes-Feldman-Weiss [4], if H is a countable amenable group and Y is a standard Borel H-space, then for every probability measure ν on Y, there exists an H-invariant Borel subset B ⊆ Y with ν(B) = 1 such that E_{H}^{Y} ↾ B is hyperfinite. ( In [4], the result is only stated for the case when ν is H-quasi-invariant; i.e. when H.N is ν-null for every ν-null Borel subset N ⊆ Y. However, as Kechris points out in [12], the result is easily seen to hold for an arbitrary probablity measure ν. To see this, let H = {h_{n} | n ≥ 1} and consider the probability measure ν^{∗} on Y defined by

ν^{∗}(A) = ∑_{n=1}^{∞} ν(h_{n}.A)/2^{n}.

Then ν^{∗} is an H-quasi-invariant probability measure which agrees with ν on every H-invariant Borel set.) In particular, let ν = fμ be the probability measure defined on Y by ν(A) = μ(f^{−1}(A)) for each Borel subset A ⊆ Y; and let B be an Hinvariant Borel subset B with ν(B) = 1 such that E_{H}^{Y} ↾ B is hyperfinite. Let X_{0} = f^{−1}(B). Then X_{0} is a Γ-invariant subset of X with μ(X_{0}) = 1. Consider the Borel function f ↾ X_{0} : X_{0} → B. By Hjorth-Kechris [9, Theorem 10.5], since Γ is a Kazhdan group and E_{H}^{Y} ↾ B is hyperfinite, there exists a Γ-invariant Borel subset M ⊆ X_{0} with μ(M) = 1 such that f maps M into a single E_{H}^{Y}-class.

3. Torsion-free abelian groups of rank two

In this section, we shall prove Theorem 1.6. For each ℓ = 1, 2, let S_{ℓ}(ℚ^{2}) be the GL_{2}(ℚ)-invariant Borel subset consisting of the groups G ∈ S(ℚ^{2}) of rank ℓ. In the proof of Theorem 1.6, we shall quickly reduce to the case when f : X → S_{2}(ℚ^{2}). Following Król [14], our analysis of the groups G ∈ S_{2}(ℚ^{2}) will split into various cases depending on the structure of certain invariants T (G) and C(G), which we shall now describe. So let G ∈ S_{2}(ℚ^{2}) be a torsion-free abelian group of rank two. Then

T (G) = {𝔞 | There exists 0 ≠ a ∈ G such that τ(a) = 𝔞}

denotes the set of types which are realised in G. If 0 ≠ a ∈ G, then the pure subgroup of G generated by a consists of those elements c ∈ G such that there exist k, ℓ ∈ ℤ with k ≠ 0 such that kc = ℓa. Now let a, b ∈ G be a basis of G and let A, B be the pure subgroups of G generated by a, b respectively. Then G/(A + B) is a torsion group. In fact, by Król 2.23 [14],

G/(A + B) ≅ ⨁_{p∈ℙ} C(t_{p})

for some t_{p} ∈ ℕ ∪ {∞}, where

(a) if t_{p} < ∞, then C(t_{p}) is a cyclic group of order p^{t_{p}}; and

(b) if t_{p} = ∞, then C(t_{p}) is a Prüfer p-group; i.e. an infinite locally cyclic p-group. The characteristic χ of G/(A + B) is defined to be the function χ ∈ (ℕ ∪ {∞})^{ℙ} such that χ(p) = t_{p}; and we say that the basis a, b of G realises the triple

⟨𝔞, 𝔟, χ⟩,

where 𝔞 = τ(a) and 𝔟 = τ(b). We define C(G) to be the set of all such triples which are realised by some basis of G.

Lemma 3.1. Suppose that G ∈ S_{2}(ℚ^{2}) and that 𝔞, 𝔟 ∈ T (G) are two fixed (not necessarily distinct) types. If χ_{1}, χ_{2} are characteristics such that ⟨𝔞, 𝔟, χ_{1}⟩, ⟨𝔞, 𝔟, χ_{2}⟩ ∈ C(G), then χ_{1} and χ_{2} belong to the same type.

Proof. This is an immediate consequence of Theorems 2.25, 2.27 and 2.28 of Król [14].

Lemma 3.2. Let 𝔞, 𝔟 be two fixed (not necessarily distinct) types. Then there only exist countably many groups G ∈ S_{2}(ℚ^{2}) such that ⟨𝔞, 𝔟, χ⟩ ∈ C(G) for some characteristic χ belonging to the zero-type. (Here the zero-type is the type which contain the characteristic ϑ ∈ (ℕ ∪ {∞})^{ℙ} such that ϑ(p) = 0 for all p ∈ ℙ.)

Proof. Let A, B be torsion-free abelian groups of rank one such that τ(A) = 𝔞 and τ(B) = 𝔟. Then each such group G can be realised up to isomorphism as an extension

0 → A ⊕ B → G → F → 0

of A ⊕ B by a suitably chosen finite abelian group F. Fix a finite abelian group F and let F = ⨁_{i=1}^{k} C_{i} be a decomposition of F into a direct sum of cyclic groups C_{i} of order m_{i}. Then by Theorems 7.14 and 7.17 [19],

Ext_{ℤ}(F, A ⊕ B) ≅ ∏_{i=1}^{k} Ext_{ℤ}(C_{i}, A ⊕ B)

≅ ∏_{i=1}^{k} (A/m_{i}A) ⊕ (B/m_{i}B) .

Thus Ext_{ℤ}(F, A ⊕ B) is a finite group; and the result follows easily.

If 𝔞, 𝔟 are types, then we write 𝔞 ≤ 𝔟 iff there exists characteristics χ ∈ 𝔞 and ϑ ∈ 𝔟 such that χ(p) ≤ ϑ(p) for all p ∈ ℙ. The types 𝔞 and 𝔟 are said to be comparable if either 𝔞 ≤ 𝔟 or 𝔟 ≤ 𝔞. Otherwise, 𝔞 and 𝔟 are incomparable. If 𝔞, 𝔟 are types, then 𝔞 ∧ 𝔟 is the type containing the characteristic

⟨min{χ(p), ϑ(p)} | p ∈ ℙ⟩,

where χ, ϑ are arbitrary elements of 𝔞, 𝔟 respectively.

We are now ready to begin the proof of Theorem 1.6. So let X be a standard Borel Γ-space with an invariant ergodic probablity measure μ and let f : X → S(ℚ^{2}) be a Borel function such that xE_{Γ}^{X}y implies f(x) ≅_{2} f(y). Let E = E_{Γ}^{X} and for each x ∈ X, let G_{x} = f(x) ∈ S(ℚ^{2}). Since μ is ergodic, there exists an Γ-invariant subset X_{0} with μ(X_{0}) = 1 such that one of the following two cases occurs. (A) G_{x} ∈ S_{1}(ℚ^{2}) for all x ∈ X_{0}. (B) G_{x} ∈ S_{2}(ℚ^{2}) for all x ∈ X_{0}. To simplify notation, we shall assume that X_{0} = X. (We shall make this simplifying assumption each time that we appeal to the ergodicity of μ.) First suppose that G_{x} ∈ S_{1}(ℚ^{2}) for all x ∈ X. By Theorem 1.3, ≅_{2}↾ S_{1}(ℚ^{2}) is hyperfinite; and so ≅_{2}↾ S_{1}(ℚ^{2}) is equal to E_{ℤ}^{S_{1}(ℚ^{2})} for some Borel action of ℤ on S_{1}(ℚ^{2}). By Theorem 2.4, there exists a Γ-invariant Borel subset M with μ(M) = 1 such that f maps M into a single ≅_{2}-class. Hence we can suppose that f : X → S_{2}(ℚ^{2}). Appealing to the ergodicity of μ once again, we can now suppose that one of the following three cases occurs.

(I) |T (G_{x})| > 2 for all x ∈ X. (II) |T (G_{x})| = 2 for all x ∈ X. (III) |T (G_{x})| = 1 for all x ∈ X.

TORSION-FREE ABELIAN GROUPS OF RANK TWO 13

Case (I). Suppose that |T (G_{x})| > 2 for all x ∈ X. By Król 2.5 [14], since each |T (G_{x})| > 2, it follows that each T (G_{x}) contains at least one pair of incomparable types. Let g : X → ℚ^{2} be a Borel function which selects a basis g(x) = ⟨a_{x}, b_{x}⟩ ∈ G_{x} × G_{x} such that the types 𝔞_{x} = τ(a_{x}) and 𝔟_{x} = τ(b_{x}) are incomparable. Let A_{x}, B_{x} be the pure subgroups of G_{x} generated by a_{x}, b_{x} respectively and let χ_{x} be the characteristic of the torsion group G_{x}/(A_{x} + B_{x}). Suppose that x ∈ X and that χ_{x} belongs to the zero-type. Then by Król 4.2 [14], it follows that T (G_{x}) = {𝔞_{x}, 𝔟_{x}, 𝔞_{x} ∧ 𝔟_{x}}. In particular, {𝔞_{x}, 𝔟_{x}} is the unique pair of incomparable types in T (G_{x}). Hence Lemma 3.1 implies that if xEy, then χ_{x} belongs to the zero-type iff χ_{y} belongs to the zero-type. Since μ is ergodic, we can suppose that one of the following two cases occurs.

(a) For all x ∈ X, χ_{x} belongs to the zero-type.

(b) For all x ∈ X, χ_{x} does not belong to the zero-type.

First suppose that χ_{x} belongs to the zero-type for all x ∈ X. Let

Y = (ℕ ∪ {∞})^{ℙ} × (ℕ ∪ {∞})^{ℙ}

and let h : X → Y be the Borel function defined by h(x) = ⟨χ(a_{x}), χ(b_{x})⟩. Let F be the countable Borel equivalence relation on Y such that

⟨χ(a), χ(b)⟩ F ⟨χ(c), χ(d)⟩ iff {τ(a), τ(b)} = {τ(c), τ(d)}.

Since the similarity relation ≡ on (ℕ ∪ {∞})^{ℙ} is hyperfinite, it follows easily that there exists a Borel action of the amenable group H = (ℤ × ℤ) ⋊ Sym(2) on Y such that F = E_{H}^{Y}; and clearly if xEy, then h(x)F h(y). Hence Theorem 2.4 implies that there exists an Γ-invariant subset X_{0} with μ(X_{0}) = 1 and a fixed pair of types {𝔞, 𝔟} such that {𝔞_{x}, 𝔟_{x}} = {𝔞, 𝔟} for all x ∈ X_{0}. By Lemma 3.2, the image f[X_{0}] consists of only countably many groups G ∈ S_{2}(ℚ^{2}). Hence there exists a Borel subset X_{1} with μ(X_{1}) > 0 and a fixed group G ∈ S_{2}(ℚ^{2}) such that f(x) = G for all x ∈ X_{1}. Let M = Γ.X_{1}. Since μ is ergodic, μ(M) = 1 and clearly f[M] is contained in a single ≅_{2}-class.

Hence we can suppose that for all x ∈ X, χ_{x} does not belong to the zero-type. Let g : X → P(ℙ) be the Borel map defined by

g(x) = {p ∈ ℙ | p^{m}G_{x} = G_{x} for all m ≥ 1}.

14 SIMON THOMAS

Clearly if xEy, then g(x) = g(y). By the ergodicity of μ, we can suppose that there exists a fixed set P of primes such that g(x) = P for all x ∈ X. Let R be the subgroup of the multiplicative group ℚ^{∗} consisting of those rational numbers of the form

r = ±p_{1}^{m_{1}}p_{2}^{m_{2}} … p_{k}^{m_{k}},

where p_{1}, …, p_{k} ∈ P and m_{1}, …, m_{k} ∈ ℤ; and let

D = \{( \begin{matrix}r & 0 \\ 0 & r \end{matrix}) | r ∈ R\} ⩽ GL_{2}(ℚ).

(If P = ∅, then we set R = {1, −1}.) By Król 4.3 [14], Aut(G_{x}) = D for all x ∈ X. Let

Y = {G ∈ S_{2}(ℚ^{2}) | There exists x ∈ X such that G_{x} ≅_{2} G}.

Then Y is a GL_{2}(ℚ)-invariant Borel subset of S_{2}(ℚ^{2}); and the action of GL_{2}(ℚ) on Y induces a free action of H = GL_{2}(ℚ)/D on Y. Hence we can define a Borel cocycle β : Γ × X → H by

β(π, x) = the unique φ ∈ H such that φ[G_{x}] = G_{π.x}
= the unique φ ∈ H such that φ[f(x)] = f(π.x)

Let Z be the center of GL_{2}(ℚ) and let p : H → P GL_{2}(ℚ) = GL_{2}(ℚ)/Z be the natural surjective homomorphism. Then we can define a Borel cocycle

α : Γ × X → P GL_{2}(ℚ)

by α = p ∘ β. By Theorem 2.3, there exists an equivalent cocycle

γ : Γ × X → P GL_{2}(ℚ)

such that γ[Γ × X] is contained in a finite subgroup K of P GL_{2}(ℚ). Let B : X → P GL_{2}(ℚ) be a Borel function such that:

(*) for all π ∈ Γ, α(π, x) = B(π.x)γ(π, x)B(x)^{−1} μ-a.e.

It is easily checked that if x satisfies (*) and xEy, then y also satisfies (*). To simplify notation, we shall assume that (*) holds for all x ∈ X. Clearly there exists a Borel subset X_{1} ⊆ X with μ(X_{1}) > 0 and a fixed element ψ ∈ P GL_{2}(ℚ) such

TORSION-FREE ABELIAN GROUPS OF RANK TWO 15

that B(x) = ψ for all x ∈ X_{1}. Since μ is ergodic, μ(Γ.X_{1}) = 1; and so we can also assume that X_{1} intersects every Γ-orbit on X.

Let {π_{n} | n ∈ ℕ} be a fixed enumeration of Γ with π_{0} = 1; and for each x ∈ X, let x_{1} = π_{n}.x, where n is the least integer such that π_{n}.x ∈ X_{1}. Notice that for each x ∈ X,

Ψ(x) = {α(π, x_{1}) | π.x_{1} ∈ X_{1}} ⊆ ψKψ^{−1}

is a finite subset of P GL_{2}(ℚ). Let L = Z/D so that P GL_{2}(ℚ) = H/L; and regard α(π, x_{1}) as a coset of L in H. Then β(π, x_{1}) ∈ α(π, x_{1}). It follows that

Φ(x) = {L.f(π.x_{1}) | π.x_{1} ∈ X_{1}}

is also finite of cardinality at most |K|; and it is easily checked that 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. For each x ∈ X and 1 ≤ i ≤ k − 1, let x_{i+1} = π_{n}.x_{1}, where n is the least integer such that

(i) π_{n}.x_{1} ∈ X_{1}, and (ii) f(π_{n}.x_{1}) ∉ {L.f(x_{j}) | 1 ≤ j ≤ i}. Let f̃: X → S_{2}(ℚ^{2})^{k} be the Borel function defined by

f̃(x) = ⟨f(x_{1}), …, f(x_{k})⟩.

Let F be the countable Borel equivalence relation arising from the orbits of the natural action of

W = \underbrace{(L × ⋯ × L)}_{k copies} ⋊ Sym(k)

on S_{2}(ℚ^{2})^{k}. If xEy, then Φ(x) = Φ(y) and so f̃(x)F f̃(y). Clearly W is an amenable group; and so by Theorem 2.4, there exists an Γ-invariant Borel subset M with μ(M) = 1 such that f̃ maps M into a single F -class; and this implies that f maps M into a single ≅_{2}-class. Case (II). Suppose that |T (G_{x})| = 2 for all x ∈ X. By Król 2.6 [14], the types 𝔞_{x}, 𝔟_{x} ∈ T (G_{x}) are comparable; say, 𝔞_{x} < 𝔟_{x}. Furthermore, by Król 2.26 [14], if b_{1}, b_{2} ∈ G_{x} satisfy τ(b_{1}) = τ(b_{2}) = 𝔟_{x}, then b_{1} and b_{2} are linearly dependent. It follows that there exists a Borel function g : X → GL_{2}(ℚ) such that

g(x)[\{b ∈ G_{x} | τ(b) = 𝔟_{x}\}] ⩽ \{( \begin{matrix}q \\ 0 \end{matrix} ) | q ∈ ℚ\}

16 SIMON THOMAS

for all x ∈ X. Let f′ : X → S_{2}(ℚ^{2}) be the Borel function defined by f′(x) = g(x)[G_{x}] and let G′_{x} = g(x)[G_{x}]. Let H be the subgroup consisting of the upper triangular matrices of GL_{2}(ℚ). Then H is soluble and hence is amenable. Notice that if φ ∈ GL_{2}(ℚ) satisfies φ[G′_{x}] = G′_{y}, then

φ[{b ∈ G′_{x} | τ(b) = 𝔟_{x}}] = {c ∈ G′_{y} | τ(c) = 𝔟_{y}};

and so φ has the form

φ = (\begin{matrix}r & s \\ 0 & t \end{matrix})

for some r, s, t ∈ ℚ. Thus G′_{x} ≅_{2} G′_{y} iff there exists φ ∈ H such that φ[G′_{x}] = G′_{y}. By Theorem 2.4, there exists an SL_{2}(ℤ)-invariant Borel subset M with μ(M) = 1 such that f′ maps M into a single H-orbit on S_{2}(ℚ^{2}); and this implies that f maps M into a single ≅_{2}-class. Case (III). Suppose that |T (G_{x})| = 1 for all x ∈ X; say, T (G_{x}) = {𝔞_{x}}. Arguing as in Case (I), we can suppose that there exists a fixed type 𝔞 such that 𝔞_{x} = 𝔞 for all x ∈ X. After slightly adjusting f if necessary, we can also suppose that for all x ∈ X, the standard basis elements e_{1}, e_{2} of ℚ^{2} are contained in G_{x} and that e_{1}, e_{2} realise the same characteristic in G_{x}. (With a little more effort, we could even reduce to the case when the characteristic realised by e_{1}, e_{2} in G_{x} is fixed for all x ∈ X. However, this extra uniformity is not required in the following argument.) Let A_{x}, B_{x} be the pure subgroups of G_{x} generated by e_{1}, e_{2} respectively and let χ_{x} be the characteristic of the torsion group G_{x}/(A_{x} + B_{x}). Arguing as in Case (I), we can also suppose that for all x ∈ X, χ_{x} does not belong to the zero-type.

Since 𝔞_{x} = 𝔞 for all x ∈ X, it follows that there is a fixed set P of primes such that

P = {p ∈ ℙ | p^{m}G_{x} = G_{x} for all m ≥ 1}

for all x ∈ X. Let R be the subgroup of the multiplicative group ℚ^{∗} consisting of those rational numbers of the form

r = ±p_{1}^{m_{1}}p_{2}^{m_{2}} … p_{k}^{m_{k}},

where p_{1}, …, p_{k} ∈ P and m_{1}, …, m_{k} ∈ ℤ; and let

D = \{( \begin{matrix}r & 0 \\ 0 & r \end{matrix}) | r ∈ R\} ⩽ GL_{2}(ℚ).

TORSION-FREE ABELIAN GROUPS OF RANK TWO 17

(Once again, if P = ∅, then we set R = {1, −1}.) Then D ⩽ Aut(G_{x}) for all x ∈ X. By the ergodicity of μ, we can suppose that one of the following two cases occurs.

(a) D = Aut(G_{x}) for all x ∈ X.

(b) D is a proper subgroup of Aut(G_{x}) for all x ∈ X.

First suppose that D = Aut(G_{x}) for all x ∈ X; and let

Y = {G ∈ S_{2}(ℚ^{2}) | There exists x ∈ X such that G_{x} ≅_{2} G}.

Then the action of GL_{2}(ℚ) on Y induces a free action of H = GL_{2}(ℚ)/D on Y; and hence we can define a Borel cocycle β : Γ × X → H by

β(π, x) = the unique φ ∈ H such that φ[G_{x}] = G_{π.x}.

Arguing as in Case (I), we see that there exists an Γ-invariant Borel subset M with μ(M) = 1 such that f maps M into a single ≅_{2}-class.

Hence we can suppose that D is a proper subgroup of Aut(G_{x}) for all x ∈ X. Fix some x ∈ X and let

φ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Aut(G_{x}) ∖ D.

By Król 4.8 [14], bc ≠ 0 and \sqrt{d} ∉ ℚ, where

d = (a + d)^{2} − 4(ad − bc).

Let QE(G_{x}) be the ring of quasi-endomorphisms of G_{x}; i.e. QE(G_{x}) consists of those linear transformations η ∈ Mat_{2}(ℚ) such that there exists an integer m ≠ 0 with mη ∈ End(G_{x}), the endomorphism ring of G_{x}. By Król 2.29 [14], since the characteristic χ_{x} of the torsion group G_{x}/(A_{x} + B_{x}) does not belong to the zerotype, it follows that G_{x} is not decomposable into a direct sum of rank one subgroups. Hence by Król 5.10 [14], QE(G_{x}) is the ring S_{x} of all matrices η ∈ Mat_{2}(ℚ) such that there exists a rational number q ∈ ℚ and an arbitrary scalar matrix λ such that

η = qφ + λ.

Furthermore, by Król 5.11 [14], S_{x} is isomorphic to the quadratic field extension ℚ(\sqrt{d}) via the map which sends the identity matrix to 1 ∈ ℚ and sends φ to \sqrt{d}.

18 SIMON THOMAS

In particular, there are only countably many possibilities for QE(G_{x}); and so there exists a Borel subset X_{1} with μ(X_{1}) > 0 and a fixed subring S of Mat_{2}(ℚ) such that QE(G_{x}) = S for all x ∈ X_{1}. By the ergodicity of μ, we can suppose that X = Γ.X_{1}; and after slightly adjusting f if necessary, we can suppose that QE(G_{x}) = S for all x ∈ X. Thus if x, y ∈ X, then G_{x} ≅_{2} G_{y} iff there exists ψ ∈ N = N_{GL_{2}(ℚ)}(S) such that ψ[G_{x}] = G_{y}. Hence by Theorem 2.4, it is enough to show that N is an amenable group. To see this, let S ≅ ℚ(\sqrt{d}), where \sqrt{d} ∉ ℚ. Since Aut(S) ≅ Aut(ℚ(\sqrt{d})) is cyclic of order 2, it follows that the centraliser C = C_{N}(S) satisfies [N : C] ≤ 2. By the Double Centraliser Theorem for finite dimensional central simple algebras,

dim_{ℚ} S . dim_{ℚ} C_{Mat_{2}(ℚ)}(S) = dim_{ℚ} Mat_{2}(ℚ).

(For example, see Section 12.7 [18].) It follows that C_{Mat_{2}(ℚ)}(S) = S and hence C is the multiplicative group of the field S. Consequently N is abelian-by-finite and so N is amenable.

This completes the proof of Theorem 1.6.

4. A cocycle reduction result

In this section, we shall prove Theorem 2.3. But first we need to recall some notions from valuation theory. (A clear account of this material can be found in Margulis [17, Chapter I].) Let F be an algebraic number field; i.e. a finite extension of the field ℚ of rational numbers. Let R be the set of all non-equivalent valuations of F and let R_{∞} ⊂ R be the set of archimedean valuations. For each ν ∈ R, let F_{ν} be the completion of F relative to ν. If ν ∈ R_{∞}, then F_{ν} = ℝ or F_{ν} = ℂ; and if ν ∈ R ∖ R_{∞}, then F_{ν} is a totally disconnected local field; i.e. a finite extension of the field ℚ_{p} of p-adic numbers for some prime p.

Let S ⊆ R be a set of valuations of F. Then an element x ∈ F is said to be S-integral iff |x|_{ν} ≤ 1 for each non-archimedean valuation ν ∉ S. The set of all Sintegral elements is a subring of F, which will be denoted by F (S). Furthermore, F is the union of the subrings F (S), where S ranges over the finite sets of valuations of the field F. For any S ⊆ R, we define

GL_{n}(F (S)) = {φ ∈ GL_{n}(F) | The entries of φ and φ^{−1} belong to F (S)};

TORSION-FREE ABELIAN GROUPS OF RANK TWO 19

and for each F -subgroup H(F) of GL_{n}(F), we define H(F (S)) = H(F)∩GL_{n}(F (S)).

Theorem 2.3 is a straightforward consequence of the following theorem, which collects together results of Zimmer [25] and Adams-Spatzier [2].

Theorem 4.1. Let Γ be a countable Kazhdan group and let X be a standard Borel Γ-space with an invariant ergodic probability measure μ. Suppose that α : Γ × X → P SL_{2}(K) is a Borel cocycle, where either K = ℝ or K is a totally disconnected local field. Then there exists an equivalent cocycle γ such that γ[Γ×X] is contained in a compact subgroup of P SL_{2}(K).

Proof. First suppose that K = ℝ. By Zimmer [25, Theorem 10], α is equivalent to a cocycle γ such that

γ[Γ × X] ⊆ H ⩽ P SL_{2}(ℝ),

where H is an algebraic subgroup of P SL_{2}(ℝ) such that H is a Kazhdan group; and Corollary 19 [25] implies that every Kazhdan subgroup of P SL_{2}(ℝ) is compact.

Now suppose that K is a totally disconnected local field. By Serre [20], there is an action of P SL_{2}(K) as a group of automorphisms of a countable simplicial tree T such that the stabilizer of each point of T is compact; and by Adams-Spatzier [2, Theorem 1.1], α is equivalent to a cocycle γ such that γ[Γ × X] is contained in the stabilizer of some point of T.

We are now ready to begin the proof of Theorem 2.3. So let Γ be a countable Kazhdan group and let X be a standard Borel Γ-space with an invariant ergodic probablity measure μ. Suppose that α : Γ × X → P GL_{2}(ℚ) is a Borel cocycle. After passing to a finite ergodic extension of X if necessary, we can suppose that α[Γ × X] ⊆ P GL_{2}^{+}(ℚ), the subgroup of P GL_{2}(ℚ) of index 2 corresponding to the matrices φ ∈ GL_{2}(ℚ) such that det(φ) > 0. (For example, see Propositions 2.5 and 2.6 [1].) Since Γ is a Kazhdan group, Lemma 2.2 [26] implies that α is equivalent to a cocycle β taking values in a finitely generated subgroup Λ of P GL_{2}^{+}(ℚ). So there exists a finite set {p_{1}, …, p_{n}} of primes and a finite set S of valuations of the algebraic number field F = ℚ[\sqrt{p_{1}}, …, \sqrt{p_{n}}] such that Λ ⩽ P SL_{2}(F (S)). Clearly we can suppose that S contains the set R_{∞} of archimedean valuations of F. It follows that if P SL_{2}(F (S)) is identified with its image under the diagonal embedding into

G_{S} = ∏_{ν∈S} P SL_{2}(F_{ν}),

then P SL_{2}(F (S)) is a discrete subgroup of G_{S}. (For example, see Section I.3.2 [17].) Of course, F_{ν} is a totally disconnected local field for each ν ∈ S ∖ R_{∞}; and since F is a totally real field, it follows that F_{ν} = ℝ for all ν ∈ R_{∞}. For each ν ∈ S, let p_{ν} : G_{S} → P SL_{2}(F_{ν}) be the canonical projection; and, viewing β as a cocycle into G_{S}, let β_{ν} : Γ × X → P SL_{2}(F_{ν}) be the Borel cocycle defined by β_{ν} = p_{ν} ∘ β. By Theorem 4.1, β_{ν} is equivalent to a cocycle taking values in a compact subgroup H_{ν} of P SL_{2}(F_{ν}). It follows that β is equivalent to a cocycle taking values in the compact subgroup H = ∏_{ν∈S} H_{ν} of G_{S}. By Adams-Kechris [1, Proposition 2.4], there exists g ∈ G_{S} and a cocycle γ : Γ × X → Λ such that β ∼ γ and γ takes values in the finite subgroup Λ ∩ gHg^{−1} of P GL_{2}(ℚ). This completes the proof of Theorem 2.3.

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