THE CLASSIFICATION PROBLEM FOR p-LOCAL TORSION-FREE ABELIAN GROUPS OF RANK TWO

GREG HJORTH AND SIMON THOMAS

Abstract. We prove that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are incomparable with respect to Borel reducibility.

1. Introduction

This paper is a contribution to the project [4, 3, 1, 8, 11, 12, 14] of trying to understand the Borel complexity of the classification problem for the torsion-free abelian groups of finite rank. Let R(ℚ^{n}) be the standard Borel space consisting of the additive subgroups of ℚ^{n} of rank n and let ≅_{n} be the isomorphism relation on R(ℚ^{n}). Also, for each prime p, let R^{p}(ℚ^{n}) be the standard Borel space consisting of the p-local subgroups of ℚ^{n} of rank n and let ≅_{n}^{p} be the isomorphism relation on R^{p}(ℚ^{n}). (Recall that an abelian group A is said to be p-local iff A is q-divisible for every prime q ≠ p.)

In 1998, Hjorth [3] proved that (≅_{1}) <_{B} (≅_{2}). Two years later, making essential use of earlier work of Adams-Kechris [1], Thomas [13] proved that (≅_{n}) <_{B} (≅_{n+1}) for all n ≥ 2. In the same paper, Thomas also proved that if p is a fixed prime, then (≅_{n}^{p}) <_{B} (≅_{n+1}^{p}) for all n ≥ 1. However, this left open the natural problem of whether the isomorphism relations on R^{p}(ℚ^{n}) and R^{q}(ℚ^{n}) were comparable with respect to Borel reducibility when p ≠ q and n ≥ 2. (It is trivial that (≅_{1}^{p}) ∼_{B} (≅_{1}^{q}) for any primes p ≠ q, since there are only two p-local groups of rank 1 up to isomorphism.) In 2002, Thomas [14] proved that if p ≠ q and n ≥ 3, then ≅_{n}^{p} and ≅_{n}^{q} are incomparable with respect to Borel reducibility. However, his proof made essential use of the fact that if n ≥ 3, then SL_{n}(ℤ) is a Kazhdan group. Consequently, the problem remained open in the case when n = 2.

Research partially supported by NSF Grants.

2 GREG HJORTH AND SIMON THOMAS

In this paper, we shall prove that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are also incomparable with respect to Borel reducibility. Theorem 1.1. If p ≠ q are distinct primes, then ≅_{2}^{p} and ≅_{2}^{q} are incomparable with respect to Borel reducibility.

In Thomas [13, 5.7], it was shown that E_{0} <_{B} (≅_{2}^{p}) ≤_{B} (≅_{2}). However, it remained an open problem whether (≅_{2}^{p}) <_{B} (≅_{2}). In fact, it was not known whether there existed a Borel equivalence relation F such that E_{0} <_{B} F <_{B} (≅_{2}). Corollary 1.2. If p is a prime, then E_{0} <_{B} (≅_{2}^{p}) <_{B} (≅_{2}).

□

For each prime p, let E^{p} be the orbit equivalence relation induced by the action of P GL_{2}(ℚ) on the projective line P G(1, ℚ_{p}) over the field ℚ_{p} of p-adic numbers. By Thomas [13, 5.7], we have that (≅_{2}^{p}) ∼_{B} E^{p}. Hence Theorem 1.1 is an immediate consequence of the following result. Theorem 1.3. If p ≠ q, then E^{p} and E^{q} are incomparable with respect to Borel reducibility.

The analogous result for the higher dimensional projective spaces over the various p-adic fields was proved in Thomas [14], once again making use of the fact that if n ≥ 3, then SL_{n}(ℤ) is a Kazhdan group. In this paper, the use of the Kazhdan property will be replaced by a systematic use of E_{0}-ergodicity. (Most applications of the Kazhdan property exploit the incompatibility of the actions of Kazhdan groups and amenable groups; and, to a large extent, this incompatibility is captured by the notion of E_{0}-ergodicity. However, we will see a less obvious application of E_{0}-ergodicity in the proof of Theorem 3.2.) It is perhaps worth mentioning that the proof of the E_{0}-ergodicity of the relevant actions ultimately relies on the fact that SL_{2}(ℤ) satisfies a weak form of the Kazhdan property; namely, SL_{2}(ℤ) has Property (τ) with respect to its family of congruence subgroups. (See Lubotzky [9, 4.3.3.D] and Thomas [15, Section 5].)

This paper is organised as follows. In Section 2, we shall recall some basic notions and results concerning Borel equivalence relations, ergodic theory and irreducible lattices in products of real and p-adic algebraic groups. In Section 3, we shall prove a cocycle reduction theorem for the action of P SL_{2}(ℤ[1/q]) on P G(1, ℚ_{p}),

p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK 3

q ≠ p, modulo a corresponding cocycle reduction theorem for suitable actions of products of lcsc groups, which we shall prove in Section 6. Then, in Section 4, we shall generalize the results of Section 3 to finite ergodic extensions of P G(1, ℚ_{p}). Finally, in Section 5, we shall present the proof of Theorem 1.3.

2. Preliminaries

In this section, we shall recall some basic notions and results concerning Borel equivalence relations, ergodic theory and irreducible lattices in products of real and p-adic algebraic groups.

2.1. Actions of irreducible lattices. Let p be a prime. Then ℚ_{p} denotes the field of p-adic numbers and ℤ_{p} denotes the ring of p-adic integers. The projective line P G(1, ℚ_{p}) is the space of 1-dimensional subspaces of the 2-dimensional vector space V (2, ℚ_{p}) over ℚ_{p}. It is easily checked that the compact group P SL_{2}(ℤ_{p}) acts transitively on P G(1, ℚ_{p}). (For example, see Thomas [13, 6.1].) Fix some point x ∈ P G(1, ℚ_{p}) and let L be the stabilizer of x in P SL_{2}(ℤ_{p}). Then L is a closed subgroup of P SL_{2}(ℤ_{p}) and we can identify P G(1, ℚ_{p}) with the coset space P SL_{2}(ℤ_{p})/L. Throughout this paper, μ_{p} will denote the corresponding Haar probability measure on P G(1, ℚ_{p}) = P SL_{2}(ℤ_{p})/L.

Let q be a prime such that q ≠ p and let Λ = P SL_{2}(ℤ[1/q]). Then Λ is a dense subgroup of P SL_{2}(ℤ_{p}) and hence Λ acts ergodically on (P G(1, ℚ_{p}), μ_{p}). Furthermore, the ergodic Λ-space (P G(1, ℚ_{p}), μ_{p}) has discrete spectrum; i.e. the associated unitary representation of Λ on L^{2}(P G(1, ℚ_{p}), μ_{p}) is a direct sum of mutually orthogonal finite-dimensional irreducible representations. (For example, see the proof of Thomas [15, 5.7].) Now let

H = P SL_{2}(ℝ) × P SL_{2}(ℚ_{q}).

By Margulis [10, IX.1.7(iii)], if we identify Λ with its image under the diagonal embedding into H, then Λ is an irreducible lattice in H. Consider the action of H on the induced space P G(1, ℚ_{p}) × H/Λ. Let ν be the Haar probability measure on H/Λ and let μ̂_{p} = μ_{p} × ν. Then μ̂_{p} is H-invariant and H acts ergodically on (P G(1, ℚ_{p}) × H/Λ, μ̂_{p}). (For example, see Adams-Kechris [1, Section 2].) In fact, a much stronger result holds.

Lemma 2.1. (P G(1, ℚ_{p}) × H/Λ, μ̂_{p}) is an irreducible H-space; i.e. the subgroups P SL_{2}(ℝ) and P SL_{2}(ℚ_{q}) also act ergodically on (P G(1, ℚ_{p}) × H/Λ, μ̂_{p}).

Proof. Since Λ is an irreducible lattice in H and the ergodic Λ-space (P G(1, ℚ_{p}), μ_{p}) has discrete spectrum, we can argue exactly as in the proof of Zimmer [19, 2.4]. □

Remark 2.2. Let Γ = SL_{2}(ℤ[1/q]). Towards the end of the proof of Theorem 1.3, it will be necessary to regard (P G(1, ℚ_{p}), μ_{p}) as a Γ-space rather than a Λ-space. In this case, Γ is an irreducible lattice in G = SL_{2}(ℝ)×SL_{2}(ℚ_{q}); and exactly the same argument shows that G acts irreducibly on the induced space P G(1, ℚ_{p}) × G/Γ.

2.2. Borel equivalence relations and E_{0}-ergodicity. Throughout this paper, our notation follows that of Jackson-Kechris-Louveau [6] and Hjorth-Kechris [5]. In particular, suppose that G is a lcsc group; i.e. 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; and the corresponding G-orbit equivalence relation on X is denoted by E_{G}^{X}.

Suppose that E, F are Borel equivalence relations on the standard Borel spaces X, Y respectively and that μ is a (not necessarily finite) E-invariant probability measure on X. Then (E, μ) is said to be F -ergodic iff for every Borel homomorphism f : X → Y from E to F, there exists a μ-conull Borel subset M ⊆ X such that f maps M into a single F -class. In this case, for simplicity, we shall often say that E is F -ergodic. Throughout this paper, we shall mostly be concerned with E_{0}-ergodicity, where E_{0} is the Vitali equivalence relation, defined on 2^{ℕ} by xE_{0}y iff x(n) = y(n) for almost all n. Theorem 2.3. Let G be a lcsc group and let X be a standard Borel G-space with G-invariant ergodic probability measure μ. Then the following conditions are equivalent:

(I) E_{G} is not E_{0}-ergodic.

(II) There exist δ_{1}, δ_{2} with 0 < δ_{1} < δ_{2} < 1 such that for all compact K ⊆ G and all ε > 0, there exists a measurable A ⊆ X with δ_{1} < μ(A) < δ_{2} such that

ν({g ∈ K : μ(A △ g(A)) > ε}) < ε.

p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK 5

(III) There exist δ_{1}, δ_{2} with 0 < δ_{1} < δ_{2} < 1 and a sequence of measurable subsets (A_{m})_{m∈ℕ} of X, with δ_{1} < μ(A_{m}) < δ_{2} for all m, such that for any g ∈ G

μ({x ∈ A_{m} : g · x ∉ A_{m}}) → 0 as m → ∞.

Proof. This is a routine modification of results from Jones-Schmidt [7]. (Alternatively, see Appendix A of Hjorth-Kechris [5]). □

Remark 2.4. In the statement of Theorem 2.3, Condition (II) can be strengthened to:

(II)′ Let 0 < c < 1 be arbitrary. Then for all compact K ⊆ G and all ε > 0, there exists a Borel subset A ⊆ X with μ(A) = c such that

ν({g ∈ K : μ(A △ g(A)) > ε}) < ε.

3. A cocycle reduction result

The following cocycle reduction theorem, which will play a key role in the proof of Theorem 1.3, was conjectured by Thomas [15] and soon afterwards proved by Hjorth. Theorem 3.1. Let p ≠ q be distinct primes and let D be a countable group. Then for every cocycle α : P SL_{2}(ℤ[1/q]) × P G(1, ℚ_{p}) → D, there exists an equivalent cocycle β such that β(P SL_{2}(ℤ[1/q])×P G(1, ℚ_{p})) is contained in a finitely generated subgroup of D.

In the remainder of this section, we shall explain how to deduce Theorem 3.1 from the following general cocycle reduction theorem for suitable actions of products of lcsc groups, which we shall prove in Section 6. Theorem 3.2. Let G = G_{1}×G_{2} be a product of lcsc groups and let X be a standard Borel G-space with G-invariant probability measure μ. Suppose that the following conditions hold:

(a) The action of G on X is essentially free; i.e. there exists a μ-conull Ginvariant Borel subset X_{0} ⊆ X such that G acts freely on X_{0}.

(b) The action of G_{1} on X is E_{0}-ergodic.

(c) There exists an element σ ∈ G_{2} such that the transformation x ↦ σ · x is ergodic on X.

(d) G_{1} and G_{2} are compactly generated.

6 GREG HJORTH AND SIMON THOMAS

Then whenever D is a countable group and α : G×X → D is a cocycle, there exists an equivalent cocycle β such that β(G × X) is contained in a finitely generated subgroup of D.

Throughout this section, let Λ = P SL_{2}(ℤ[1/q]) and H = P SL_{2}(ℝ)×P SL_{2}(ℚ_{q}). Recall that Λ is a uniform lattice in H. Most of our effort will go into verifying that the induced action of H on P G(1, ℚ_{p}) × H/Λ satisfies the hypotheses of Theorem 3.2. Lemma 3.3. Let G = G_{1} × G_{2} be a product of lcsc groups. Suppose that Γ is a lattice in G and that S is a standard Borel Γ-space with Γ-invariant probability measure.

If the product action of Γ on S × G_{2} is E_{0}-ergodic, then the induced action of G_{1} on S × G/Γ is E_{0}-ergodic. Remark 3.4. Here we understand that Γ acts on G_{2} via the transformations

g ↦^{γ} π(γ)g, γ ∈ Γ, g ∈ G,

where π : Γ → G_{2} is the projection into the second factor of G = G_{1} × G_{2}; and this is the sense in which we can talk about a product action of Γ on S × G_{2}. Similarly, in the following proof, we can also consider the product action of G on (S × G/Γ) × G_{2}.

Proof. Suppose that the action of Γ on S × G_{2} is E_{0}-ergodic. Consider the actions of G_{1} on S × G/Γ and G on S × G/Γ × G_{2}. First note that the function

θ : S × G/Γ → S × G/Γ × G_{2},

defined by

(s, gΓ) ↦ (s, gΓ, 1_{G_{2}}),

is a Borel reduction from E_{G_{1}}^{S×G/Γ} to E_{G}^{S×G/Γ×G_{2}}; and that if A is a Borel subset of S × G/Γ, then A is conull in S × G/Γ iff the saturation of θ(A) is conull in S × G/Γ × G_{2}. It follows that E_{G_{1}}^{S×G/Γ} is E_{0}-ergodic if and only if E_{G}^{S×G/Γ×G_{2}} is E_{0}-ergodic.

Next note that S × {Γ} × G_{2} is a complete section for E_{G}^{S×G/Γ×G_{2}}, which comes equipped with the product measure on S × G_{2}; and that if B is a Borel subset of S × {Γ} × G_{2}, then B is conull in S × {Γ} × G_{2} iff the saturation of B is conull

p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK 7

in S × G/Γ × G_{2}. Furthermore, the complete section S × {Γ} × G_{2} is Γ-invariant and the action of Γ on S × {Γ} × G_{2} is isomorphic to the action of Γ on S × G_{2}. Hence the conclusion of the lemma follows from the E_{0}-ergodicity of the action of Γ on S × G_{2}. □

Lemma 3.5. The action of P SL_{2}(ℤ) on P G(1, ℚ_{p}) × P SL_{2}(ℤ_{q}) is E_{0}-ergodic.

Proof. Recall that P SL_{2}(ℤ_{p}) acts transitively on P G(1, ℚ_{p}). Hence the result is a special case of Thomas [15, 6.2]. □

Lemma 3.6. The action of P SL_{2}(ℤ[1/q]) on P G(1, ℚ_{p})×P SL_{2}(ℚ_{q}) is E_{0}-ergodic.

Proof. Let Ω = P G(1, ℚ_{p}) × P SL_{2}(ℚ_{q}). Notice that Ω_{0} = P G(1, ℚ_{p}) × P SL_{2}(ℤ_{q}) is a complete section for the action of P SL_{2}(ℤ[1/q]) on Ω; and that if A ⊆ Ω is P SL_{2}(ℤ[1/q])-invariant, then A is a conull subset of Ω iff A ∩ Ω_{0} is a conull subset of Ω_{0}. Furthermore, since

P SL_{2}(ℤ[1/q]) ∩ P SL_{2}(ℤ_{q}) = P SL_{2}(ℤ),

it follows that if x, y ∈ Ω_{0}, then x, y lie in the same P SL_{2}(ℤ[1/q])-orbit on Ω iff there exists an element γ ∈ P SL_{2}(ℤ) such that γ · x = y. Hence the result follows from Lemma 3.5. □

Lemma 3.7. (a) The action of P SL_{2}(ℝ) on P G(1, ℚ_{p}) × H/Λ is E_{0}-ergodic. (b) The action of P SL_{2}(ℚ_{q}) on P G(1, ℚ_{p}) × H/Λ is strongly mixing.

Proof. Clause 3.7(a) is an immediate consequence of Lemmas 3.6 and 3.3. By Lemma 2.1, H acts irreducibly on the induced space P G(1, ℚ_{p}) × H/Λ. In particular, P SL_{2}(ℚ_{q}) acts ergodically on P G(1, ℚ_{p}) × H/Λ. By Zimmer [19, Chapter 10], every ergodic action of P SL_{2}(ℚ_{q}) is strongly mixing and hence Clause 3.7(b) also holds. □

Proof of Theorem 3.1. Let D be a countable group and let

α : P SL_{2}(ℤ[1/q]) × P G(1, ℚ_{p}) → D

be a cocycle. By Thomas [15, 3.8], the action of Λ = P SL_{2}(ℤ[1/q]) on P G(1, ℚ_{p}) is essentially free. Hence the action of H = P SL_{2}(ℝ) × P SL_{2}(ℚ_{q}) on the induced space P G(1, ℚ_{p}) × H/Λ is also essentially free. Let

α̂ : H × P G(1, ℚ_{p}) × H/Λ → D

be the corresponding cocycle induced from α. By Lemma 3.7(a), the action of P SL_{2}(ℝ) on P G(1, ℚ_{p}) × H/Λ is E_{0}-ergodic. Also, by Lemma 3.7(b), the action of P SL_{2}(ℚ_{q}) on P G(1, ℚ_{p}) × H/Λ is strongly mixing. Hence, if we let σ ∈ P SL_{2}(ℚ_{q}) be the element corresponding to the matrix

(\begin{matrix}q & 0 \\ 0 & 1/q \end{matrix}) ∈ SL_{2}(ℚ_{q}),

then the transformation x ↦ σ · x is ergodic on P G(1, ℚ_{p}) × H/Λ. Thus the hypotheses of Theorem 3.2 are satisfied and so α̂ is equivalent to a cocycle β̂ taking values in a finitely generated subgroup of D_{0} of D. By Adams-Kechris [1, 2.3], α is also equivalent to a cocycle β taking values in D_{0}. □

4. Finite ergodic extensions

In this section, we shall first generalize the results of Section 3 to finite ergodic extensions of (P G(1, ℚ_{p}, μ_{p}); and then we shall prove a result about the quotients of such finite ergodic extensions, which generalizes Thomas [15, 6.4].

Suppose that G is a lcsc group and that X is a standard Borel G-space with invariant probability measure μ. Then a finite extension of (X, μ) consists of a standard Borel G-space X̃ with invariant probability measure μ̃, together with a Borel map π : X̃ → X, such that the following conditions are satisfied:

(i) π(g · x̃) = g · π(x̃) for all g ∈ G and x̃ ∈ X̃.

(ii) π is finite-to-one.

(iii) π_{∗}μ̃ = μ. If α : G × X → D is a cocycle, then its lift to X̃ is the cocycle α̃ : G × X̃ → D defined by α̃(g, x̃) = α(g, π(x̃)). Example 4.1. Let C be a finite nonempty set and let β : G × X → Sym C be a cocycle. Then we can define an action of G on X × C by

g · (x, c) = (g · x, β(g, x) · c).

p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK 9

Let ν be the counting probability measure on C. Then the finite extension corresponding to β consists of the standard G-Borel space X̃ = X × C with invariant probability measure μ̃ = μ × ν, together with the Borel map π(x, c) = x.

In fact, if (X, μ) is an ergodic G-space and (X̃, μ̃) is a finite ergodic extension, then it is easily checked that, after deleting null subsets if necessary, (X̃, μ̃) is isomorphic to one of the standard finite extensions described in Example 4.1. Lemma 4.2. Suppose that G is a lcsc group, X is a standard Borel space with G-invariant ergodic probability measure μ and (X̃, μ̃) is a finite ergodic extension. If G acts E_{0}-ergodically on (X, μ), then G also acts E_{0}-ergodically on (X̃, μ̃).

Proof. We can suppose that X̃ = X ×C is a standard finite extension. Let |C| = n. Then for each Borel subset B ⊆ X̃, we have that

μ̃(B) ≤ μ(π(B)) ≤ nμ̃(B).

Suppose that G does not act E_{0}-ergodically on (X̃, μ̃). Let ν be a Haar measure on G and fix some real c such that 0 < c ≤ nc < 1. By Remark 2.4, for every compact subset K ⊆ G and ε > 0, there exists a Borel subset A ⊆ X̃ with μ̃(A) = c such that

ν({g ∈ K : μ̃(A △ g · A) > ε/n}) < ε.

For all g ∈ G and A ⊆ X̃, we have that

π(A) △ g · π(A) = π(A) △ π(g · A) ⊆ π(A △ g · A);

and hence if

μ̃(A △ g · A) ≤ ε/n,

then

μ(π(A) △ g · π(A)) ≤ ε.

But this means that (X, μ) satisfies Condition 2.3(II), which is a contradiction. □

Lemma 4.3. Suppose that G = G_{1}×G_{2} is a product of lcsc groups, X is a standard Borel space with G-invariant ergodic probability measure μ and (X̃, μ̃) is a finite ergodic extension. If

(a) G_{2} acts ergodically on X, and

(b) G_{1} has no proper subgroups of finite index, then G_{2} also acts ergodically on X̃.

Proof. Once again, we can suppose that X̃ = X × C is a standard finite extension. Let ν be the counting probability measure on C. Suppose that A ⊆ X × C is a G_{2}-invariant Borel subset such that (μ × ν)(A) > 0. Then π(A) is a G_{2}-invariant Borel subset of X and μ(π(A)) > 0. Since G_{2} acts ergodically on X, it follows that μ(π(A)) = 1, and we can suppose that there exists a μ-measure 1 Borel subset B ⊆ X and an integer 1 ≤ n ≤ |C| such that A = ⋃_{b∈B}{b} × S_{b}, where each S_{b} ⊆ C has cardinality n. Furthermore, we can suppose that A has been chosen so that n is minimal. Now let g ∈ G_{1}. Then g(A) satisfies the same properties. Since g(A) ∩ A is G_{2}-invariant, it follows that either g(A) = A or g(A) ∩ A = ∅ modulo (μ × ν)-null sets. Consequently, modulo (μ × ν)-null sets,

S = {g(A) : g ∈ G_{1}}

is a finite collection of Borel sets which is permuted by the action of G_{1}. Hence there exists a subgroup G′_{1} of finite index in G_{1} which fixes each set in this collection. Since G_{1} has no proper subgroups of finite index, this implies that G′_{1} = G_{1} and S = {A}. As G = G_{1} × G_{2} acts ergodically on X × C, it follows that (μ × ν)(A) = 1. □

Next suppose that H is a lcsc group and that Λ is a lattice in H. Let X be a standard Borel Λ-space with invariant ergodic probability measure μ. Let X̃ = X × C be a finite ergodic extension of (X, μ) corresponding to a cocycle β : Λ × X → Sym(C). Also let X̂ = X × H/Λ be the induced H-space and let β̂: H × X̂ → Sym(C) be the corresponding cocycle induced from β. Consider the actions of H on the following two H-spaces:

• the induced space X̃ × H/Λ = (X × C) × H/Λ; and

• the finite extension X̂ × C = (X × H/Λ) × C corresponding to the cocycle

β̂. By simply unwrapping the definitions, we easily obtain the following result.

Lemma 4.4. With the above hypotheses, (X × C) × H/Λ and (X × H/Λ) × C are isomorphic H-spaces via the measure-preserving map (x, c, hΓ) ↦ (x, hΓ, c).

□

p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK 11

We can now easily generalize the results of Section 3 to finite ergodic extensions. For the rest of this section, let p ≠ q be distinct primes and let (X̃, P SL_{2}(ℤ[1/q]), μ̃) be a finite ergodic extension of (P G(1, ℚ_{p}), P SL_{2}(ℤ[1/q]), μ_{p}). Once again, let Λ = P SL_{2}(ℤ[1/q]) and H = P SL_{2}(ℝ) × P SL_{2}(ℚ_{q}). Lemma 4.5. (a) The action of P SL_{2}(ℝ) on X̃ × H/Λ is E_{0}-ergodic.

(b) The action of P SL_{2}(ℚ_{q}) on X̃ × H/Λ is strongly mixing.

Proof. Once again, we can suppose that X̃ = P G(1, ℚ_{p}) × C is a standard finite extension. Let Ω = P G(1, ℚ_{p}) × H/Λ. By Lemma 4.4, the H-space X̃ × H/Λ is isomorphic to the finite ergodic extension Ω × C of Ω. Hence Clause 4.5(a) is an immediate consequence of Lemmas 3.7(a) and 4.2. Similarly, by Lemmas 3.7(b) and 4.3, P SL_{2}(ℚ_{q}) acts ergodically on X̃ × H/Λ. Since every ergodic action of P SL_{2}(ℚ_{q}) is strongly mixing, it follows that Clause 4.5(b) also holds. □

Arguing exactly as in the proof of Theorem 3.1, we now obtain the following result. (Note that the action of Λ = P SL_{2}(ℤ[1/q]) on the finite ergodic extension X̃ is also essentially free.) Theorem 4.6. If D is a countable group and α : P SL_{2}(ℤ[1/q]) × X̃ → D is a cocycle, then there exists an equivalent cocycle β such that β(P SL_{2}(ℤ[1/q]) × X̃) is contained in a finitely generated subgroup of D.

□

Before we can state the final result of this section, we need to introduce the notion of a quotient of a Borel action. Definition 4.7. For each i = 1, 2, let Γ_{i} be a countable group and let X_{i} be a standard Borel Γ_{i}-space with an invariant ergodic probability measure μ_{i}. Then (X_{2}, Γ_{2}, μ_{2}) is a quotient of (X_{1}, Γ_{1}, μ_{1}) iff there exist

(i) a surjective homomorphism φ : Γ_{1} → Γ_{2} and (ii) a Borel function f : X_{1} → X_{2}

such that the following conditions are satisfied:

(a) f_{∗}μ_{1} = μ_{2}; and

(b) for each γ ∈ Γ_{1}, f(γ · x) = φ(γ) · f(x) for μ_{1}-a.e. x ∈ X_{1}.

In other words, (X_{2}, Γ_{2}, μ_{2}) is isomorphic to a factor of (X_{1}, Γ_{1}, μ_{1}).

12 GREG HJORTH AND SIMON THOMAS

In the statement of the following result, ℙ denotes the set of primes and ℚ_{∞} denotes the field ℝ of real numbers. For purely technical reasons, we shall regard (X̃, μ̃) as an SL_{2}(ℤ[1/q])-space rather than as a P SL_{2}(ℤ[1/q])-space. Lemma 4.8. Suppose that A is a finite nonempty subset of ℙ ∪ {∞} and that H_{A} = ∏_{p∈A} P SL_{2}(ℚ_{p}). Let M be a proper closed subgroup of H_{A} such that H_{A}/M has finite volume and let m_{A} be the Haar probability measure on H_{A}/M. Let Λ be a lattice in H_{A} and let Λ^{+} be a countable group such that Λ ⩽ Λ^{+} ⩽ H_{A}.

Then (H_{A}/M, Λ^{+}, m_{A}) is not a quotient of (X̃, SL_{2}(ℤ[1/q]), μ̃).

Proof. Otherwise, the proof of Thomas [15, 6.4] shows that (X̃, SL_{2}(ℤ[1/q]), μ̃) has a quotient of the form (H_{C}/Δ_{C}, Λ′, m_{C}), where:

(i) H_{C} = ∏_{p∈C} P SL_{2}(ℚ_{p}) for some nonempty subset C ⊆ A; (ii) Δ_{C} is an irreducible lattice in H_{C} and m_{C} is the Haar probability measure

on H_{C}/Δ_{C}; and (iii) the subgroup Λ′ has non-compact closure in H_{C}.

By Zimmer [19, Chapter 10], the action of H_{C} on (H_{C}/Δ_{C}, m_{C}) is strongly mixing. Hence, letting π denote the associated unitary representation of H_{C} on L^{2}(H_{C}/Δ_{C}, m_{C}), we have that for all f, g ∈ L^{2}(H_{C}/Δ_{C}, m_{C}), the matrix coefficient ⟨π(h) · f, g⟩ → 0 as h → ∞ in H_{C}. Since Λ′ has non-compact closure in H_{C}, there exists a sequence ⟨γ_{n} : n ≥ 0⟩ of elements of Λ′ such that γ_{n} → ∞ in H_{C} as n → ∞. It follows that the action of Λ′ on (H_{C}/Δ_{C}, m_{C}) is weakly mixing. (For example, see the argument in Bekka-Mayer [2, 2.15(iii)].) Since (P G(1, ℚ_{p}), SL_{2}(ℤ[1/q]), μ_{p}) has discrete spectrum, it follows that the finite ergodic extension (X̃, SL_{2}(ℤ[1/q]), μ̃) has generalized discrete spectrum. (For a clear account of actions with generalized discrete spectrum, see Zimmer [17, Section 4].) But this contradicts Zimmer [16, 9.10], which implies that an ergodic action with generalized discrete spectrum has no weakly mixing factors. □

5. The proof of Theorem 1.3

In this section, we shall prove Theorem 1.3. For each prime p, let E^{p} be the orbit equivalence relation induced by the action of P GL_{2}(ℚ) on P G(1, ℚ_{p}). Suppose that p ≠ q are distinct primes and that f : P G(1, ℚ_{p}) → P G(1, ℚ_{q}) is a Borel reduction from E^{p} to E^{q}. Let Λ = P SL_{2}(ℤ[1/q]) and let E be the orbit equivalence relation

p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK 13

induced by the action of Λ on P G(1, ℚ_{p}). Then f is an ℵ_{0}-to-1 Borel homomorphism from E to E^{q}. In the remainder of the proof, we shall show that no such Borel homomorphism exists. Let

Y = {y ∈ P G(1, ℚ_{q}) : g · y ≠ y for all 1 ≠ g ∈ P GL_{2}(ℚ)}.

In other words, Y is the Borel subset of P G(1, ℚ_{q}) on which P GL_{2}(ℚ) acts freely. Arguing as in Thomas [15, 3.8], we see that P G(1, ℚ_{q})∖Y is a countable set. Hence we can suppose that f(x) ∈ Y for all x ∈ P G(1, ℚ_{p}).

Let α : Λ × P G(1, ℚ_{p}) → P GL_{2}(ℚ) be the Borel cocycle defined by

α(g, x) · f(x) = f(g · x)

for all g ∈ Λ and x ∈ P G(1, ℚ_{p}). For each matrix M ∈ GL_{2}(ℚ), let [M] be the corresponding element of P GL_{2}(ℚ). Let A = ℚ^{∗}/(ℚ^{∗})^{2} and let ψ : P GL_{2}(ℚ) → A be the homomorphism defined by

ψ(g) = det(M_{g})(ℚ^{∗})^{2},

where M_{g} ∈ GL_{2}(ℚ) satisfies [M_{g}] = g. Then ker ψ = P SL_{2}(ℚ). Consider the cocycle ψ ∘α : Λ×P G(1, ℚ_{p}) → A. By Theorem 3.1, ψ ∘α is equivalent to a cocycle taking values in a finitely generated subgroup F ⩽ A. It easily follows that α is equivalent to a cocycle α′ taking values in L = ψ^{−1}(F). (For example, see the proof of Adams-Kechris [1, 6.1].) Since A is an abelian group of exponent 2, it follows that F is a finite subgroup of A and thus [L : P SL_{2}(ℚ)] < ∞. Hence by Adams-Kechris [1, 2.5], there exists a finite ergodic extension (X̃, μ̃) of (P G(1, ℚ_{p}), μ_{p}) such that the lift α̃ : Λ × X̃ → L of α′ is equivalent to a cocycle β taking values in P SL_{2}(ℚ). Let f̃: X̃ → Y be the corresponding Borel function such that for all g ∈ Λ,

β(g, x) · f̃(x) = f̃(g · x) for μ̃-a.e. x ∈ X̃.

Then f̃is an ℵ_{0}-to-1 Borel homomorphism to E^{q} from the orbit equivalence relation Ẽ induced by the action of Λ on X̃. Lemma 5.1. Ẽ is E_{0}-ergodic.

Proof. By Thomas [15, 6.1], Λ acts E_{0}-ergodically on (P G(1, ℚ_{p}), μ_{p}). Hence the result follows from Lemma 4.2. □

14 GREG HJORTH AND SIMON THOMAS

By Theorem 4.6, β is equivalent to a cocycle β′ taking values in a finitely generated subgroup of P SL_{2}(ℚ). To simplify the notation, we shall suppose that β′ = β. So there exists a finite nonempty set of primes T = {q_{1}, …, q_{t}} such that β takes values in Λ_{T} = P SL_{2}(ℤ[1/q_{1}, …, 1/q_{t}]). Of course, this means that f̃ is also an ℵ_{0}-to-1 Borel homomorphism from Ẽ to the Λ_{T} -orbit equivalence relation on Y. Lemma 5.2. There does not exist a μ̃-measure 1 subset of X̃ which is mapped by f̃ into a single Λ_{T} -orbit.

□

Let Γ = SL_{2}(ℤ[1/q]) and G = SL_{2}(ℝ) × SL_{2}(ℚ_{q}). For the rest of the proof, we shall regard (X̃, μ̃) as a Γ-space rather than a Λ-space. Of course, Ẽ is also the Γ-orbit equivalence relation on X̃. Let X̂ = X̃ × G/Γ be the induced G-space and let μ̂ = μ̃ × ν, where ν is the Haar probability measure on G/Γ. Lemma 5.3. (X̂, μ̂) is an irreducible G-space.

Proof. Recall that, by Lemma 4.4, X̂ is isomorphic to a finite ergodic extension of the induced G-space P G(1, ℚ_{p}) × G/Γ. By Remark 2.2, G acts irreducibly on P G(1, ℚ_{p}) × G/Γ. Applying Lemma 4.3 to both factors of G, we see that G also acts irreducibly on X̂. □

Putting together Lemmas 4.8, 5.1, 5.2 and 5.3, we now see that the hypotheses of Thomas [15, 8.2] are satisfied. Hence q ∈ T = {q_{1}, ⋯, q_{t}} and there exist

(i) a subgroup Γ_{0} of finite index in Γ, (ii) a Γ_{0}-invariant Borel subset X_{0} ⊆ X̃ with μ̃(X_{0}) > 0, (iii) a homomorphism φ̄ : Γ_{0} → Λ_{T}, and (iv) a Borel map f̄: X_{0} → Y

such that the following conditions hold:

(a) φ̄(Γ_{0}) is a subgroup of finite index in P SL_{2}(ℤ[1/q]).

(b) f̄(γ · x) = φ̄(γ) · f̄(x) for all γ ∈ Γ_{0} and x ∈ X_{0}.

Let μ̃_{0} be the Γ_{0}-invariant probability measure on X_{0} defined by

μ̃_{0}(Z) = μ̃(Z)/μ̃(X_{0}).

Then f̄_{∗}μ̃_{0} is a φ̄(Γ_{0})-invariant probability measure on P G(1, ℚ_{q}). By clause (a), φ̄(Γ_{0}) is a dense subgroup of P SL_{2}(ℚ_{q}) and hence f̄_{∗}μ̃_{0} is P SL_{2}(ℚ_{q})-invariant.

p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK 15

However, this is a contradiction, since it is well-known that there are no P SL_{2}(ℚ_{q})- invariant probability measures on P G(1, ℚ_{q}). This completes the proof of Theorem 1.3. Remark 5.4. Examining the previous argument, we see that a rather stronger result holds; namely, under a suitable interpretation, if p ≠ q are distinct primes, then E^{p} is E^{q}-ergodic. In more detail, consider the action of P GL_{2}(ℚ) on P G(1, ℚ_{p}). Although there does not exist a P GL_{2}(ℚ)-invariant measure on P G(1, ℚ_{p}), the probability measure μ_{p} is quasi-invariant; i.e. for all Borel subsets A ⊆ P G(1, ℚ_{p}) and all elements γ ∈ P GL_{2}(ℚ), we have that μ_{p}(γ(A)) = 0 iff μ_{p}(A) = 0. In other words, there is a P SL_{2}(ℚ)-invariant notion of a null or conull subset of P G(1, ℚ_{p}). Hence it is natural to say that if E is a Borel equivalence relation on a standard Borel space Y, then E^{p} is E-ergodic iff for every Borel homomorphism f : P G(1, ℚ_{p}) → Y from E^{p} to E, there exists a μ_{p}-conull subset of P G(1, ℚ_{p}) which is mapped by f into a single E-class. With this interpretation, applying Thomas [15, 8.9], we also see that if F is a treeable countable Borel equivalence relation on a standard Borel space Z, then E^{p} is F -ergodic.

6. The proof of Theorem 3.2

In this section, we shall prove Theorem 3.2. First we need the following entirely abstract result. Definition 6.1. Let G be a lcsc group and let X be a standard Borel G-space with G-invariant probability measure μ. Suppose that F is a Borel equivalence relation on X. Then the orbit equivalence relation E_{G}^{X} is said to be included in F almost everywhere, written

E_{G}^{X} ⊆ F μ-a.e.,

iff for all g ∈ G, we have that

xF g · x for μ-a.e. x ∈ X.

Lemma 6.2. Let H = H_{1} × H_{2} be a product of lcsc groups and let X be a standard Borel H-space with H-invariant probability measure μ. Suppose the following conditions hold:

(a) There exists an increasing sequence of Borel equivalence relations

F_{0} ⊆ F_{1} ⊆ ⋯ ⊆ F_{n} ⊆ ⋯

on X such that E_{H}^{X} = ⋃_{n} F_{n}.

(b) E_{H_{2}}^{X} ⊆ F_{0} μ-a.e.

(c) H_{2} acts ergodically.

(d) H_{1} is compactly generated. Then there exists an integer N ≥ 0 such that E_{H}^{X} ⊆ F_{N} μ-a.e.

Proof. Throughout this proof, let ν be a Haar measure on H_{1}. First note that for each n ∈ ℕ, σ_{1} ∈ H_{1} and σ_{2} ∈ H_{2}, the set

A_{σ_{1}}^{n} = {x ∈ X : xF_{n}σ_{1} · x}

is σ_{2}-invariant on a μ-conull subset of X; and hence by ergodicity of the H_{2}-action, each A_{σ_{1}}^{n} is either μ-null or μ-conull. In particular, if for each n, we let

H_{1}^{n} = {σ ∈ H_{1} : xF_{n}σ · x for μ-a.e. x ∈ X}
= {σ ∈ H_{1} : A_{σ}^{n} is μ-conull},

then (H_{1}^{n}) is an increasing sequence of ν-measurable subgroups such that

H_{1} = ⋃_{n∈ℕ} H_{1}^{n}.

Note that if ν(H_{1}^{n}) > 0, then H_{1}^{n} must actually be an open subgroup of H_{1}. Hence, after possibly discounting a few initial terms, we obtain a representation of H_{1} as an increasing union of open subgroups. Since H_{1} is compactly generated, there exists an integer N ≥ 0 such that H_{1}^{N} = H_{1}, which means that E_{H_{1}}^{X} ⊆ F_{N} μ-a.e. Since E_{H_{2}}^{X} ⊆ F_{0} μ-a.e., it follows that E_{H}^{X} ⊆ F_{N} μ-a.e. □

We can are now ready to begin the proof of Theorem 3.2. Let G = G_{1} × G_{2} be a product of lcsc groups and let X be a standard Borel G-space with G-invariant probability measure μ. Assume that the hypotheses of Theorem 3.2 are satisfied. Let D be a countable group and let α : G × X → D be a cocycle. In order to simplify notation, we shall assume that G acts freely on X.

p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK 17

Let D = ⋃_{n≥0} D_{n} be a representation of D as an increasing union of finitely generated subgroups. Define an increasing sequence of Borel equivalence relations

E_{0} ⊆ E_{1} ⊆ ⋯ ⊆ E_{n} ⊆ ⋯

on X by setting

• xE_{n}y iff xE_{G}^{X}y and α(g, x) ∈ D_{n}, where g ∈ G is the unique element such

that g · x = y.

Clearly E_{G}^{X} = ⋃_{n≥0} E_{n}. Let E_{σ} be the equivalence relation induced by the ergodic transformation T_{σ} : X → X.

First, if there exists a natural number m such that E_{σ} ⊆ E_{m} μ-a.e., then Theorem 3.2 follows easily by two applications of Lemma 6.2. Hence we can suppose that no such natural number exists.

Until further notice, fix some natural number m. Let Y be the Borel subset of X consisting of the points x such that

lim inf_{N→∞} \frac{|\{n ∈ [−N, N] : xE_{m}T_{σ}^{n}(x)\}|}{2N + 1} > 1/2,

Suppose that μ(Y) > 0. Since the transformation x ↦ σ · x is ergodic, it follows that Y intersects the E_{σ}-class [x]_{E_{σ}} for μ-a.e. x ∈ X. Let F be the equivalence relation on X defined by xF y iff xE_{σ}y and xE_{m}y. Then it is easily checked that each nonempty intersection [x]_{E_{σ}} ∩ Y consists of a single F -class. Hence we can replace α by an equivalent cocycle β such that for μ-a.e. x ∈ X, β(σ^{n}, x) ∈ D_{m} for all n ∈ ℤ. But this puts us into the case which we have already dealt with in the previous paragraph. Thus we can suppose that μ(Y) = 0.

By the Birkhoff Ergodic Theorem, if A ⊆ X is a Borel subset, then

lim_{N→∞} \frac{|\{n ∈ [−N, N] : T_{σ}^{n}(x) ∈ A\}|}{2N + 1} = μ(A) for μ-a.e. x ∈ X.

Hence, since μ(Y) = 0, if μ(A) > 1/2, then for μ-a.e. x ∈ X, there exists an integer n such that

x¬E_{m}T_{σ}^{n}(x) and T_{σ}^{n}(x) ∈ A.

Therefore, by a process of exhaustion, we may find disjoint measurable B, C ⊆ X, each of a measure 1/4, together with partitions

B = ⨆_{n∈ℤ} B_{n} and C = ⨆_{n∈ℤ} C_{n},

18 GREG HJORTH AND SIMON THOMAS

such that • T_{σ}^{n}[B_{n}] = C_{n} for each n, and

• x¬E_{m}T_{σ}^{n}(x) for all x ∈ B_{n}.

Claim 6.3. There exists a natural number integer n ≥ 1 such that

μ(\{x ∈ X : T_{σ}^{n}(x)E_{m}x\}) < \frac{4}{5}.

Proof of Claim 6.3. Fix some large N_{0} such that μ(⋃_{|n|<N_{0}} B_{n}) > \frac{1}{4} − \frac{1}{100}. Let B̂ = ⋃_{|n|<N_{0}} B_{n} and Ĉ = ⋃_{|n|<N_{0}} C_{n}. Suppose that |n| < N_{0} and x ∈ B_{n}. Consider an integer ℓ ∈ ℤ such that xE_{m}T_{σ}^{ℓ}(x). Since x ∈ B_{n}, it follows that x¬E_{m}T_{σ}^{n}(x) and so T_{σ}^{ℓ}(x)¬E_{m}T_{σ}^{n}(x). Thus we obtain that ℓ satisfies T_{σ}^{ℓ−n}(T_{σ}^{n}(x))¬E_{m}T_{σ}^{n}(x). Hence, for any N ≥ N_{0}, we obtain that

|{ℓ : |ℓ| < N, T_{σ}^{ℓ}(x)E_{m}x}| ≤ |{ℓ : |ℓ| < N − N_{0}, T_{σ}^{ℓ−n}(T_{σ}^{n}(x))¬E_{m}T_{σ}^{n}(x)}| + 2N_{0}.

Since this is true for every x ∈ B_{n}, we obtain that

∫_{B_{n}}(x ↦ |{ℓ : |ℓ| < N, T_{σ}^{ℓ}(x)E_{m}x}|)dμ

is bounded above by

∫_{B_{n}}(x ↦ |{ℓ : |ℓ| < N − N_{0}, T_{σ}^{ℓ−n}(T_{σ}^{n}(x))¬E_{m}T_{σ}^{n}(x)}|)dμ + 2N_{0}μ(B_{n}).

Using that the fact that T_{σ}^{n} is measure preserving and replacing the variable x with y = T_{σ}^{n}(x), we obtain that

∫_{B_{n}}(x ↦ |{ℓ : |ℓ| < N, T_{σ}^{ℓ}(x)E_{m}x}|)dμ

≤ ∫_{C_{n}}(y ↦ |{ℓ : |ℓ| < N − N_{0}, T_{σ}^{ℓ−n}(y)¬E_{m}y}|)dμ + 2N_{0}μ(C_{n}),

which, in turn, after replacing ℓ by the variable ℓ′ = ℓ − n, is bounded by

≤ ∫_{C_{n}}(y ↦ |{ℓ′ : |ℓ′| < N, T_{σ}^{ℓ′}(y)¬E_{m}y}|)dμ + 2N_{0}μ(C_{n}).

Interchanging the order of the summand and the integral, we obtain that

∑_{{ℓ:|ℓ|<N}} μ({x ∈ B_{n} : T_{σ}^{ℓ}(x)E_{m}x})

≤ 2N_{0}μ(C_{n}) + ∑_{{ℓ:|ℓ|<N}} μ({y ∈ C_{n} : T_{σ}^{ℓ}(y)¬E_{m}y}).

p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK 19

Summing over n with |n| < N_{0}, we get that

∑_{{ℓ:|ℓ|<N}} ∑_{{n:|n|<N_{0}}} μ({x ∈ B_{n} : T_{σ}^{ℓ}(x)E_{m}x})

≤ 2N_{0}μ( ⋃_{|n|<N_{0}} C_{n}) + ∑_{{ℓ:|ℓ|<N}} ∑_{{n:|n|<N_{0}}} μ({y ∈ C_{n} : T_{σ}^{ℓ}(y)¬E_{m}y}),

and hence, by the definition of B̂ and Ĉ,

∑_{{ℓ:|ℓ|<N}} μ({x ∈ B̂ : T_{σ}^{ℓ}(x)E_{m}x}) ≤ 2N_{0} + ∑_{{ℓ:|ℓ|<N}} μ({y ∈ Ĉ : T_{σ}^{ℓ}(y)¬E_{m}y}).

Now since Ĉ ⊆ C and μ(B \ B̂) < \frac{1}{100}, we obtain that

∑_{{ℓ:|ℓ|<N}} μ({x ∈ B : T_{σ}^{ℓ}(x)E_{m}x})

≤ 2N_{0} + \frac{2N − 1}{100} + ∑_{\{ℓ:|ℓ|<N\}} μ(\{y ∈ C : T_{σ}^{ℓ}(y)¬E_{m}y\}).

Essentially the same argument also gives the following inequality:

∑_{{ℓ:|ℓ|<N}} μ({x ∈ C : T_{σ}^{ℓ}(x)E_{m}x})

≤ 2N_{0} + \frac{2N − 1}{100} + ∑_{\{ℓ:|ℓ|<N\}} μ(\{y ∈ B : T_{σ}^{ℓ}(y)¬E_{m}y\}).

Combining these inequalities together, we have that

∑_{{ℓ:|ℓ|<N}} μ({x ∈ B ∪ C : T_{σ}^{ℓ}(x)E_{m}x})

≤ 4N_{0} + \frac{4N − 2}{100} + ∑_{\{ℓ:|ℓ|<N\}} μ(\{y ∈ B ∪ C : T_{σ}^{ℓ}(y)¬E_{m}y\}).

Thus for large N,

∑_{{ℓ:|ℓ|<N}} μ({x ∈ B ∪ C : T_{σ}^{ℓ}(x)E_{m}x})

≤ \frac{5N}{100} + ∑_{\{ℓ:|ℓ|<N\}} μ(\{y ∈ B ∪ C : T_{σ}^{ℓ}(y)¬E_{m}y\}).

Thus, in particular, there exists an integer ℓ with |ℓ| < N such that

μ(\{x ∈ B ∪ C : T_{σ}^{ℓ}(x)E_{m}x\}) < \frac{5}{100} + μ(\{y ∈ B ∪ C : T_{σ}^{ℓ}(y)¬E_{m}y\}).

20 GREG HJORTH AND SIMON THOMAS

Since

μ({x ∈ B ∪ C : T_{σ}^{ℓ}(x)E_{m}x}) = μ(B ∪ C) − μ({x ∈ B ∪ C : T_{σ}^{ℓ}(x)¬E_{m}x})

= \frac{1}{2} − μ(\{x ∈ B ∪ C : T_{σ}^{ℓ}(x)¬E_{m}x\}),

it follows that

\frac{1}{2} − μ(\{x ∈ B ∪ C : T_{σ}^{ℓ}(x)¬E_{m}x\}) < \frac{5}{100} + μ(\{x ∈ B ∪ C : T_{σ}^{ℓ}(x)¬E_{m}x\}),

and hence that

μ(\{x ∈ B ∪ C : T_{σ}^{ℓ}(x)¬E_{m}x\}) > \frac{1}{4} − \frac{5}{200} > \frac{1}{5}.

Finally, to see that we can choose ℓ ≥ 1, note that

μ({x ∈ X : T_{σ}^{ℓ}(x)E_{m}x}) = μ({x ∈ X : T_{σ}^{−ℓ}(x)E_{m}x}).

This completes the proof of the Claim 6.3. □

Having established this claim, for each m, let N_{m} ≥ 1 be the least natural number such that

μ(\{x ∈ X : T_{σ}^{N_{m}}(x)E_{m}x\}) < \frac{4}{5} = \frac{8}{10}.

Now note that for all sufficiently large m, we have that

μ(\{x ∈ X : T_{σ}(x)E_{m}x\}) > \frac{9}{10};

and hence, for such sufficiently large m, it follows that

\frac{7}{10} < μ(\{x ∈ X : T_{σ}^{N_{m}}(x)E_{m}x\}) < \frac{8}{10}.

For each m, let

A_{m} = {x ∈ X : T_{σ}^{N_{m}}(x)¬E_{m}x}

be the complementary set, and note that the above inequality gives that

\frac{3}{10} > μ(A_{m}) > \frac{2}{10}

for all sufficiently large m. Claim 6.4. For each g ∈ G_{1},

μ({x ∈ A_{m} : g · x ∉ A_{m}}) → 0 as m → ∞.

p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK 21

Proof of Claim 6.4. Fix some g ∈ G_{1}. For each m ≥ 0, let

D_{m} = {x ∈ X : g · x¬E_{m}x or g · (T_{σ}^{N_{m}}(x))¬E_{m}T_{σ}^{N_{m}}(x)}.

As m → ∞, we have that

μ({x ∈ X : g · x¬E_{m}x}) → 0

and hence μ(D_{m}) → 0. Thus it is enough to show that for all m ≥ 0,

g(A_{m} ∖ D_{m}) ⊆ A_{m}.

Let x ∈ A_{m} ∖ D_{m}. Then, by the definition of D_{m}, we have that

g · (T_{σ}^{N_{m}}(x))E_{m}T_{σ}^{N_{m}}(x);

and hence, since the actions of T_{σ}^{N_{M}} and g commute,

T_{σ}^{N_{m}}(g · x)E_{m}T_{σ}^{N_{m}}(x).

The assumption that x ∈ A_{m} ∖ D_{m} also implies that

g · xE_{m}x and T_{σ}^{N_{m}}(x)¬E_{m}x.

It follows that

T_{σ}^{N_{m}}(g · x)¬E_{m}g · x

and hence g · x ∈ A_{m}. This completes the proof of Claim 6.4. □

Thus (A_{m})_{m∈ℕ} is a sequence of measurable sets, almost invariant under the action of G_{1}, such that 2/10 < μ(A_{m}) < 3/10 for all m ∈ ℕ. But, by Theorem 2.3, this contradicts the hypothesis that G_{1} acts E_{0}-ergodically on X. This completes the proof of Theorem 3.2.

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Mathematics Department, UCLA, Los Angeles, CA 90095

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