The classification of torsion-free abelian groups of finite rank up to isomorphism and up to quasi-isomorphism

The isomorphism and quasi-isomorphism relations on the $p$-local torsion-free abelian groups of rank $n\geq3$ are incomparable with respect to Borel reducibility.


INTRODUCTION
This paper extends some recent work (by Hjorth, Kechris, Adams and Thomas) on the complexity of the classification problem for torsion-free abelian groups of finite rank. In 1937, Baer solved the classification problem for torsion-free abelian groups of rank 1. The rank 2 groups resisted satisfactory classification for sixty years, after which Hjorth [Hjo] used the theory of countable Borel equivalence relations to prove that the classification problem for rank 2 torsion-free abelian groups is genuinely more complex than that for rank 1 torsion-free abelian groups. Building upon the work of Adams-Kechris [AK], Thomas [Tho2] later proved that the complexity of the classification problem for torsion-free abelian groups of rank n strictly increases with n.
As a stepping stone to this result, Thomas [Tho2] proved that the classification problem for torsion-free abelian groups of rank n up to quasi-isomorphism strictly increases in complexity with n. Here, A and B are said to be quasi-isomorphic iff A is commensurable with an isomorphic copy of B. This left open the question of which of the two classification problems for torsion-free abelian groups of fixed rank n is more complex: that up to isomorphism or that up to quasi-isomorphism. In this paper, we prove that if n ≥ 3 then the two problems have incomparable complexities.
In order to state these results more formally, we must use the terminology of Borel equivalence relations. The idea, due to Friedman-Stanley [FS] and Hjorth-Kechris [HK], is that often a "classification problem" may be regarded as an equivalence relation on a standard Borel space. (A standard Borel space is a separable, completely metrizable space equipped only with its σ-algebra of Borel sets.) For instance, any torsion-free abelian This is a part of the author's doctoral thesis, which was written under the supervision of Simon Thomas. This work was partially supported by NSF grant DMS 0600940. group of rank n is isomorphic to a subgroup of Q n . Hence, any torsion-free abelian group of rank n can be presented as an element of the standard Borel space R(n) of all subgroups of Q n that contain a basis of Q n . Studying the complexity of the classification problem for torsion-free abelian groups of rank n thus amounts to studying the complexity of the isomorphism relation ∼ = n on R(n).
The equivalence relations ∼ = n lie in the class of countable Borel equivalence relations, which we now describe. An equivalence relation E on a standard Borel space X is called Borel iff it is a Borel subset of the product space X × X, and it is called countable iff all of its equivalence classes are countable. If Γ is a countable group of Borel bijections of X, then the corresponding orbit equivalence relation defined by x E Γ y ⇐⇒ x, y lie in the same Γ-orbit is easily seen to be a countable Borel equivalence relation. For instance, it is easy to verify that subgroups A, B ≤ Q n are isomorphic iff there exists g ∈ GL n (Q) such that B = g(A). Hence, ∼ = n is the orbit equivalence relation induced by the action of GL n (Q) on R(n), and thus it is a countable Borel equivalence relation.
In fact, by Feldman and Moore [FM], any countable Borel equivalence relation on a standard Borel space X arises as the orbit equivalence relation induced by a Borel action of a countable group Γ on X. We remark that neither the group Γ nor its action is canonically determined by E Γ ; the case of ∼ = n is special in the sense that there is a natural group action which induces it.
We now discuss how to compare the complexity of two equivalence relations. If E, F are equivalence relations on the standard Borel spaces X, Y, respectively, then we write E ≤ B F and say that E is Borel reducible to F iff there exists a Borel function f : The relationship E ≤ B F means that elements of X can be explicitly classified up to E using invariants from the quotient space Y/F, considered with its quotient Borel structure.
Additionally, E ≤ B F implies that structurally, X/E is a simpler space of invariants than Y/F.
This was the first naturally occurring example of an infinite ≤ B -chain. Shortly after this was found, Thomas found an infinite ≤ B -antichain, again consisting of the isomorphism relations on various spaces of torsion-free abelian groups of finite rank. If p is prime, the abelian group A is said to be p-local iff it is infinitely q-divisible for every prime q = p. Let R(n, p) denote the subspace of R(n) consisting of just the p-local torsion-free abelian groups of rank n, and let ∼ = n,p denote the restriction of ∼ = n to R(n, p). Thomas's theorem says that if n ≥ 3 and p, q are distinct primes, then ∼ = n,p is Borel incomparable with ∼ = n,q . (This was later extended to include the case n = 2 by Hjorth-Thomas [HT].) In our comparison of isomorphism and quasi-isomorphism, we shall consider only the p-local groups. It will actually be necessary to restrict our attention to a slightly smaller space, that of p-local torsion-free abelian groups of fixed divisible rank. Here, the divisible rank of a (finite rank) torsion-free abelian group A is defined as the maximum k of the ranks of the divisible quotients of A. We let R(n, p, k) denote the subspace of R(n, p) consisting of just the p-local torsion-free abelian groups of rank n and of divisible rank k. Let ∼ = k n,p denote the restriction of ∼ = n to R(n, p, k).
Theorem A. Let n ≥ 3 and p a prime, and suppose that k, l < n and k = l. Then ∼ = k n,p is Borel incomparable with ∼ = l n,p .
We now turn to a comparison of the isomorphism and quasi-isomorphism relations on R (n, p, k). Recall that torsion-free abelian groups A, B ≤ Q n are said to be quasi-isomorphic iff A is commensurable with an isomorphic copy of B (i.e., there exists B ′ ∼ = B such that A ∩ B ′ has finite index in A and in B ′ ). Let ∼ k n,p denote the quasi-isomorphism equivalence relation on R(n, p, k). Thomas found the quasi-isomorphism relation easier to work with in [Tho2], for reasons which will become clear later on in this paper. However, the next theorem shows that the classification of (p-local) torsion-free abelian groups up to quasiisomorphism is not simpler than that up to isomorphism.
It should be noted that by Theorem 4.4 of [Tho1], when k = n − 1 the notion of quasiisomorphism coincides with that of isomorphism.
It follows easily from Theorems A and B, together with [Tho1,Theorem 4.7] that for n ≥ 3, the isomorphism and quasi-isomorphism relations on the space of all local (that is, p-local for some p) torsion-free abelian groups of rank n are also incomparable.
Conjecture. For n ≥ 3, the isomorphism and quasi-isomorphism relations on the space R(n) of all torsion-free abelian groups of rank n are Borel incomparable. This paper is organized as follows. In the second section, we introduce ergodic theory and homogeneous spaces. Our main example is the k-Grassmann space Gr k Q n p of k-dimensional subspaces of the n-dimensional vector space Q n p over the p-adics. We then prove Theorem 2.4, which gives a characterization of certain action-preserving maps between the k-Grassmann spaces. In the third section, we state a cocycle superrigidity theorem of Adrian Ioana, and derive Corollary 3.3, which roughly states that a homomorphism of SL n (Z)-orbits between Grassmann spaces is a slight perturbation of an action-preserving map. We then combine 2.4 and 3.3 to prove the auxiliary result that for l = n − k, the orbit equivalence relations induced by the action of GL n (Q) on Gr k Q n p and on Gr l Q n p are Borel incomparable. In the fourth section, we use the Kurosh-Malcev completion of torsion-free abelian groups A ≤ Q n to establish a connection (Lemma 4.1) between the space of p-local torsion-free abelian groups and the Grassmann spaces. In the last section, we put 2.4, 3.3, and 4.1 together to prove Theorems A and B.

ERGODIC THEORY OF HOMOGENEOUS SPACES
In this section, we define the notion of ergodicity of a measure-preserving action, which plays an essential role in the theory of countable Borel equivalence relations. We then consider the case of countable groups Γ acting on homogeneous spaces for compact K such that Γ ≤ K (by "homogeneous," we simply mean that K acts transitively). As an example, we introduce the Grassmann space of k-dimensional subspaces of Q n p . The material of this section is self contained, but we shall see later that there is a close relationship between Grassmann spaces and spaces of p-local torsion-free abelian groups.
Ergodicity and Borel reductions. Let Γ be a countable group acting in a Borel fashion on the standard Borel space X. If X carries a Borel probability measure µ, then we write Γ (X, µ) to indicate that Γ acts on X in a µ-preserving fashion. (When µ is clear from context, we often write Γ X.) As before, we let E Γ denote the orbit equivalence relation on X induced by the action of Γ. We say that the action Γ (X, µ) is ergodic iff every Γ-invariant subset of X is null or conull for µ. We shall use the characterization that This characterization leads to an important generalization of ergodicity which will arise in our arguments. First, if E, F be equivalence relations on standard Borel spaces X, Y, we define that a function f : (This corresponds to using Y/F as a space of incomplete invariants for the E-classification problem on X.) By the last paragraph, Γ (X, µ) is ergodic iff every Borel homomorphism from E Γ to Id Y is constant on a conull set, where Id Y represents the equality relation on the standard Borel space Y. More generally, if F is a Borel equivalence relation on Y, A countable-to-one Borel homomorphism from E to F is called a weak Borel reduction from E to F. We write E ≤ w B F if there exists a weak Borel reduction from E to F. We shall use the fact that if E, F are countable Borel equivalence relations and µ is nonatomic, then For the first implication, suppose that E is F-ergodic and f is a weak Borel reduction from E to F. Then there exists a conull subset M ⊂ dom E such that f (M) is contained in a single F-class. Since E and F are countable and f is countable-to-one, it follows that M is a countable conull set, contradicting the fact that µ is nonatomic. The second implication of (2.1) is clear from the definitions.
Ergodic components. If Γ (X, µ) is ergodic and Λ ≤ Γ is a subgroup of finite index, then there exists a partition X = Z 1 ⊔ · · · ⊔ Z N of X into Λ-invariant subsets such that for each i: The Λ-spaces Z i are called the ergodic components for the action Γ (X, µ). The set of ergodic components is determined uniquely up to null sets by the inclusion of Λ into Γ and the action of Γ on (X, µ).
Homogeneous spaces. Let K be a compact, second countable group. If K acts continuously and transitively on the standard Borel space X, then X is said to be a homogeneous space for K. Every homogeneous space for K is thus in K-preserving bijection with K/L for some closed subgroup L ≤ K. Since K/L carries a unique K-invariant probability measure (the projection of the Haar probability measure on K), it follows that X does as well. Now if Γ ≤ K is a countable dense subgroup, then the action of Γ on K/L clearly preserves the Haar measure, and moreover it is uniquely ergodic with respect to the Haar measure.
(Here, the action Γ Y is said to be uniquely ergodic iff there is a unique Γ-invariant probability measure on Y.) It is easy to see that unique ergodicity implies ergodicity, and so Γ K/L is ergodic. Next, suppose that Λ ≤ Γ is a subgroup of finite index. Then the ergodic components for the action of Λ are precisely the orbits ofΛ (the closure in K) on K/L and each ergodic component is again a homogeneous space for the compact group K 0 =Λ. If Λ Γ is a normal subgroup of finite index, then Γ acts as a transitive permutation group on theΛorbits, i.e., on the ergodic components for Λ. (For proofs of the last few claims, see [Tho3,Lemma 2.2].) Example: Grassmann spaces. Let n be a natural number and p a prime. Denote by Q n p the canonical n-dimensional vector space over the field of p-adic numbers. Then the k-Grassmann space of Q n p , denoted Gr k Q n p , is the set of k-dimensional subspaces of Q n p . Since the compact group SL n (Z p ) acts transitively on Gr k Q n p [Tho3, Proposition 6.1] we can view Gr k Q n p as a homogeneous SL n (Z p )-space. Accordingly, it carries a corresponding Haar probability measure and the dense subgroup SL n (Z) ≤ SL n (Z p ) acts (uniquely) ergodically on Gr k Q n p . We now describe the "principle congruence components" of the k-Grassmann space.
Recall that for any natural number m, the principal congruence subgroup Γ m SL n (Z) is defined by where the map on the right-hand side is the canonical surjection. It is easily seen that the Hence, the ergodic components of Gr k Q n p corresponding to the action of Γ m are precisely the K m -orbits (for example, see [Tho3,Lemma 2.2]). We call these the m th principle congruence components of the k-Grassmann space.
For example, any V in the K p t orbit of V 0 := Q p e 1 ⊕ · · · ⊕ Q p e k can be written as the and v is congruent to 0 modulo p t . Since a is clearly invertible, one can use column operations to suppose that a = I k . So we have where p t | v means that for each entry x of v, we have that x/p t lies in Z p .
Recall that for n ≥ 3, SL n (Z) has the congruence subgroup property, meaning that every subgroup of finite index contains a principle congruence subgroup. With this, it is easy to derive the following: 2.3. Proposition. If n ≥ 3, any ergodic component for the action of a subgroup Γ ≤ SL n (Z p ) of finite index on Gr k Q n p contains a principle congruence component.
We close this section with a characterization of the action-preserving maps between ergodic components of Grassmann spaces. In what follows, when Γ X and Λ Y, we shall use the term permutation group homomorphism for a pair 2.4. Theorem. Let n ≥ 3 and suppose that k, l ≤ n. Let Γ 0 , Γ 1 be subgroups of SL n (Z) of finite index, X 0 an ergodic component for the action of Γ 0 on Gr k Q n p , and X 1 an ergodic component for the action of Γ 1 on Gr l Q n p . Suppose that: Then l = k or l = n − k, and: almost every x ∈ X 0 , where x ⊥ denotes the orthogonal complement of x with respect to the usual dot product. (c) If l = k = n/2 then either the conclusion of (a) holds or the conclusion of (b) holds.
In the proof, we shall make use of the following well-known result.
Proof of Lemma 2.5. We first suppose that n is odd, so SL n (R) is a simple group. By [Mar,Theorem IX.5.8], the Zariski closure H of φ(Γ 0 ) in SL n (R) is semisimple. Let π i : H → H i denote the projections of H onto its simple factors. Then φ i (Γ 0 ) is Zariski dense in H i , and so by the Mostow-Margulis superrigidity theorem (see [Zim,Theorem 5 and it follows that the corresponding factor H i is actually SL n (R) itself. Hence we have that φ lifts to an automorphism Φ of SL n (R). Now, it is well known that the outer automorphism group of SL n (R) is comprised just of the identity and the involution −T. In other words, we may decompose Φ as χ h • (−T) i where χ h is conjugation by an element h of SL n (R) and i = 0 or 1. (When n is even, it may be necessary to also include conjugation by a permutation matrix r of determinant −1.) Since Φ(Γ 0 ) is again a lattice of SL n (R), we clearly have that h commensurates SL n (Z). By the proof of [Zim, Proposition 6.2.2] (the statement itself should be slightly modified to accommodate this case), there exists a ∈ R * such that ah ∈ GL n (Q). Of course, χ ah = χ h , and so the proof is complete in this case.
In the case that n is even, φ determines an embeddingφ : Γ 0 /Z(Γ 0 ) → PSL n (Z). One may then carry out the above argument inside PSL n (R) to obtain thatφ where ǫ is as required.
In the next proposition we shall use the following notation.
2.6. Proposition. Let n ≥ 3, and let V ∈ Gr k Q n p and W ∈ Gr l Q n p . Suppose that K ≤ SL n (Z p ) is a subgroup of finite index. If stab K V ⊂ stab K W then l = k and W = V.
It is well-known that H acts primitively on each k-Grassmann space, i.e., H acts transitively on Gr k Q n p and the stabilizer in H of each point of Gr k Q n p is a maximal subgroup of H. It follows immediately that we have stab H V = stab H W. Now, it is not hard to see that V is uniquely determined by stab H V and so V = W.
Proof of Theorem 2.4. In the notation of Lemma 2.5, we have that φ = ǫ • χ h • (−T) i . Since the center of SL n (Z) acts trivially on Gr l Q n p , we may suppose that ǫ is the identity map and that φ = χ h • (−T) i . Having done so, φ clearly lifts to an automorphism Φ of GL n (Q p ), again defined by the formula χ h • (−T) i . For i = 0, 1, let K i denote the closure in SL n (Z p ) of Γ i , so that X i is a homogeneous K i -space. Since Φ(K 0 ) is a compact group containing Γ 1 , we have that Φ(K 0 ) ⊃ K 1 . By the same reasoning, we have Φ −1 (K 1 ) ⊃ K 0 , and so Φ(K 0 ) = K 1 . Hence, we may define a function β : K 0 → X 1 by Hence, by ergodicity of Γ 0 K 0 , there exists t ∈ K 1 such that β(k) = tL 1 for almost all k ∈ K 0 . It follows that there exists a conull subset . Indeed, suppose that s ∈ K 0 satisfies sx = x. Then choose an element k ∈ K 0 such that kk 0 , ksk 0 ∈ K * 0 (this is possibile since K 0 is compact and hence unimodular). It follows that In the first case, we can apply Proposition 2.6 to conclude that l = k and f (x) = h −1 x. In the second case we conclude that l = n − k and f (x) = h −1 x ⊥ .

A SUPERRIGIDITY THEOREM
In this section, we describe a recent cocycle superrigidity theorem of Adrian Ioana (see Chapter 3 of [Ioa]). We then derive a corollary which will be used in the proofs of our main theorems. Familiarity with Borel cocycles is needed only to understand the statement of Ioana's theorem and the proof of the corollary. While it is probably also possible to use Zimmer's cocycle superrigidity theorem in our arguments, doing so would require a more technical approach.
For i ∈ N let Γ (X i , µ i ) and ρ i : X i+1 → X i be a factor map (i.e., a Γ-invariant measure-preserving map). Then the corresponding inverse limit is a Γ-space (X, µ) together with factor maps π i : X → X i satisfying π i = ρ i • π i+1 and the usual universal property associated with inverse limits.
3.1. Definition. If (X i , µ i ) are finite Γ-spaces (with factor maps as above), then the inverse limit (X, µ) is called a profinite Γ-space.
3.2. Theorem (Theorem 3.3.2 of [Ioa]). Suppose that Γ is a countable Kazhdan group, and let (X, µ) be a profinite Γ-space with corresponding factor maps π i : X → X i . Suppose additionally that the action Γ (X, µ) is ergodic and free. If α : Γ × X → Λ is a cocycle into an arbitrary countable group Λ, then there exists i ∈ N such that α is cohomologous to a cocycle Γ × X i → Λ. More precisely, there exists i ∈ N and a cocycle α i : This is most useful in the case when α is a cocycle corresponding to a Borel homomorphism f from E Γ to E Λ , where E Λ is some orbit equivalence relation. In this case, the theorem gives hypotheses under which f can be replaced with an action-preserving map.
Proof. Let α : Γ X → Λ be the cocycle corresponding to f . By Theorem 3.2, there exists a finite factor (X ′ , µ ′ ) of (X, µ) (denote the projection map by π), and a Borel function b : X → Λ such that the adjusted cocycle depends only on g and π(x). Choose any x 0 ∈ X ′ , and let Γ 0 be the stabilizer of x 0 in Γ. Clearly Γ 0 ≤ Γ is a subgroup of finite index, and by 3.1.2(ii) of [Ioa], X 0 := π −1 (x 0 ) is an ergodic component for the action of Γ 0 . Since π is constant on X 0 , the restriction of α ′ to Γ 0 × X 0 is independent of x ∈ X 0 . It follows that φ(γ) := α ′ (γ, ·) defines a homomorphism Γ 0 → Λ, and then letting f ′ = b f it is easily seen that (φ, f ′ ) satisfies our requirements. (Of course, since α ′ need only satisfy the cocycle identity almost everywhere, one may need to delete a null set of X 0 .) We first use this corollary to establish the following Borel incomparability result.
3.4. Definition. Fix n ∈ N and p prime. Let E k SL n Z and E k GL n Q denote the orbit equivalence relations induced on Gr k Q n p by SL n (Z) and GL n (Q), respectively.
3.5. Theorem. Suppose that k, l < n and l is neither k nor n − k. Then E k GL n Q is Borel incomparable with E l GL n Q .
In the next section, we shall see that this theorem implies that the quasi-isomorphism relations on R(n, p, k), R(n, p, l) are Borel incomparable. We prove slightly more: 3.6. Theorem. Let n ≥ 4 and k, l < n, and suppose that l is neither k nor n − k. Then E k SL n Z (together with the Haar measure on Gr k Q n p ) is E l GL n Q -ergodic.
Theorem 3.5 follows immediately; any Borel reduction from E k GL n Q to E l GL n Q is clearly a weak Borel reduction from E k SL n Z to E l GL n Q , and Theorem 3.6 implies that E k SL n Z ≤ w B E l GL n Q . Proof of Theorem 3.6. Let f : Gr k Q n p → Gr l Q n p be a Borel homomorphism from E k SL n Z to E l GL n Q , and suppose towards a contradiction that f does not map a conull set into a single GL n (Q)-orbit. We first reduce the analysis to a situation where the hypotheses of Corollary 3.3 hold. Since SL n (Z) acts ergodically on Gr k Q n p , we need only argue that both SL n (Z) Gr k Q n p and GL n (Q) Gr l Q n p are free actions. While neither action is literally free, it will suffice to establish the following statements: Claim (a). The action of PSL n (Z) on Gr k Q n p is almost free.
Claim (b). The function f maps a (Haar) conull set into the free part of the action of PGL n (Q) on Gr l Q n p .
Here, if the countable group Γ acts on a standard Borel space X, then we let Fr(Γ X) := {x ∈ X : 1 = g ∈ Γ =⇒ gx = x} denote the free part of the action of Γ on X. If X carries a (not necessarily Γ-invariant) probability measure, then we say that Γ X is almost free iff Fr(Γ X) is conull.
3.7. Lemma (essentially Lemma 5.1 of [Tho1]). Suppose that f : Gr k Q n p → Gr l Q n p is a Borel homomorphism from E k SL n Z to E l GL n Q . Then either f maps a conull set into a single GL n (Q)-orbit, or there exists a conull X ⊂ Gr k Q n p such that It is clear that the lemma establishes Claim (b); Claim (a) also follows by applying it in the case when l = k and f is the identity map on Gr k Q n p . (While Thomas only stated Lemma 3.7 for the special case when k = n − 1, his argument goes through without change for arbitrary k, l. Claim. We may suppose that there exists an ergodic component Γ 0 X 0 for the action SL n (Z) Gr k Q n p and a homomorphism φ : is a homomorphism of permutation groups.
In the proof, we will in fact produce a φ such that φ(Γ 0 ) is a subgroup of SL n (Z) of finite index.
Proof of claim. Using Claims (a) and (b) together, it is not difficult to see that we may apply Corollary 3.3 to suppose that there exists an ergodic componentΓ 0 X 0 for PSL n (Z) Gr k Q n p and a homomorphism of permutation groups We wish to lift φ to a map Γ 0 → GL n (Q), where Γ 0 is the preimage in SL n (Z) ofΓ 0 . First, suppose thatφ is not injective. In this case, by Margulis's theorem on normal subgroups (Theorem 8.1.2 of [Zim]), the kernel ofφ has finite index inΓ 0 . Hence,φ has finite image and so passing to an ergodic subcomponent, we can suppose without loss of generality thatφ = 1. This implies that f isΓ 0 -invariant and sinceΓ 0 X 0 is ergodic, f is almost constant. Hence, in this case f maps a conull set into a single GL n (Q)-orbit, which is a contradiction.
Next, suppose thatφ is injective. In this case, we shall again make use of Margulis's results. The next lemma will be used in tandem with Lemma 2.5.
3.8. Lemma. If Γ 0 ≤ SL n (Z) is a finite index subgroup and φ : Γ 0 → GL n (Q) is a homomorphism, then there exists a finite index subgroup Λ ≤ Γ 0 such that φ(Λ) ≤ SL n (Z) is a subgroup of finite index.
Proof. Since Γ 0 is Kazhdan (see Theorem 1.5 of [Lub]), we have that Γ ′ 0 := [Γ 0 , Γ 0 ] is a finite index subgroup of Γ 0 (see Corollary 1.29 of [Lub]). Now since GL n (Q)/SL n (Q) ∼ = Q × is abelian, we have that (In fact, the latter ≤ is an equality.) Hence, replacing Γ 0 by Γ ′ 0 if necessary, we may suppose without loss of generality that φ(Γ 0 ) ⊂ SL n (Q). Repeating the proof of Lemma 2.5, after slightly adjusting φ if necessary, we may suppose that it extends to an automorphism of SL n (R) (the adjustment is by ǫ, in the notation of 2.5). It follows that φ(Γ 0 ) is again a lattice of SL n (R). Since φ(Γ 0 ) ⊂ SL n (Q), by [Mar,IX.4.14] we have that φ(Γ 0 ) is commensurable with SL n (Z), and the lemma follows.
Although Lemma 3.8 has been stated so that it will be useful later on, for the present circumstances let us note that the same proof easily applies to the case of homomorphisms into PGL n (Q). In other words, replacing X 0 with a smaller ergodic component, we may suppose without loss of generality thatφ(Γ 0 ) is a subgroup of PSL n (Z) of finite index. Moreover, the proof of Lemma 2.5 shows thatφ lifts to a homomorphism φ : Γ 0 → SL n (Z). Since φ is a lifting, we easily obtain that f (γx) = φ(γ) f (x) for all γ ∈ Γ 0 , which completes the proof of the claim. ⊣ We now wish to maneuver into a situation where we can apply Theorem 2.4.

Claim.
We may suppose that f (X 0 ) ⊂ X 1 , where X 1 is an ergodic component for the action of φ(Γ 0 ) on Gr l Q n p .
Proof of claim. Let Z 1 , . . . , Z m be the ergodic components for the action of φ(Γ 0 ) on Gr l Q n p . Now, each f −1 (Z i ) is Γ 0 -invariant, and since Γ 0 X 0 is ergodic, exactly one of the f −1 (Z i ) is conull. Deleting a null subset of X 0 , we may suppose that f (X 0 ) ⊂ Z i , as desired. ⊣ Finally, we may apply Theorem 2.4 to conclude that l = k or l = n − k, contradicting our initial hypothesis. This completes the proof of Theorem 3.6.

COMPLETION OF LOCAL TORSION-FREE ABELIAN GROUPS
In this section, we recall some facts surrounding the Kurosh-Malcev completion of a p-local torsion-free abelian group. The completion map will allow us to relate the space of p-local torsion-free abelian groups of a fixed divisible rank with the k-Grassmann space Gr k Q n p . Using this idea, Thomas essentially showed [Tho2,Theorem 4.7] that the quasiisomorphism relation ∼ k n,p on R(n, p, k) is bireducible with the orbit equivalence relation E k GL n Q induced by the action of GL n (Q) or Gr k Q n p . (Together with Theorem 3.5, this implies that ∼ k n,p is Borel incomparable with ∼ l n,p for l = k, n − k.) We shall use the completion to establish the following more technical result. 4.1. Lemma. There exists an equivalence relation E k ∼ = on the space Gr k Q n p satisfying: Using this, we can already prove: 4.2. Theorem (Theorem A, case 1). Let n ≥ 4 and k, l < n, and suppose that l is neither k nor n − k. Then ∼ = k n,p is Borel incomparable with ∼ = l n,p .
Proof. By part (b) of Lemma 4.1, it suffices to prove that the relations E k ∼ = and E l ∼ = are Borel incomparable. Now, if f is a Borel reduction from E k ∼ = to E l ∼ = , then using part (a) of Lemma 4.1 we clearly have that f is a weak Borel reduction from E k SL n Z to E l GL n Q . This contradicts Theorem 3.6.
The completion map. Following [Fuc], for A ∈ R(n, p) we define the completion of A to a Z p -submodule of Q n p by In other words, Λ(A) is just the set of all Z p -linear combinations of elements of A where A is considered as a subset of Q n p . The completion map takes values in the standard Borel space M(n, p) of Z p -submodules of Q n p with Z p -rank exactly equal to n. In fact, by 93.1 and 93.5 of [Fuc], Λ is a bijection between R(n, p) and M(n, p). Now, if M ∈ M(n, p), then by 93.3 of [Fuc] M can be decomposed as a direct sum where V M is a vector subspace of Q n p and L is a free Z p -submodule of Q n p . The vector subspace V M is uniquely determined by M. There are many possible complementary submodules L, but in any case we have rank L = n − dim V.
By exercise 93.1 of [Fuc], the dimension of V Λ(A) is precisely the divisible rank of A. Letting M(n, p, k) denote the subspace of M(n, p) consisting of just those M with dim V M = k, it follows that Λ is a bijection of R(n, p, k) with M(n, p, k). Since Λ is GL n (Q)-invariant, we have that ∼ = k n,p is Borel equivalent (i.e., isomorphic via a Borel map) to the orbit equivalence relation induced by the action of GL n (Q) on M(n, p, k).
Decomposition of the space of completed groups. We now investigate the fibers of the "vector space part" map from M(n, p, k) to Gr k Q n p defined by M → V M , where V M is as in (4.3). So, fix V ∈ Gr k Q n p and let M ∈ M(n, p, k) be an arbitrary module such that V M = V. If W is any complementary subspace of V, meaning that V ∩ W = 0 and V ⊕ W = Q n p , then M can always be written (uniquely) as V ⊕ L where L ≤ W. Since L is free and rank L = dim W, the set {M ∈ M(n, p, k) : V M = V} is in bijection with the lattices of W.

Definition.
Let K be a discrete valuation field and R its ring of integers. Then a lattice of K l is a free R-submodule of K l of rank l. Equivalently, a lattice of K l is the R-span of l linearly independent elements of K l . We denote the set of lattices of K l by L(K l ).
By the discussion in [Ser, Section II.1.1], since Q is a dense subfield of Q p , the map L → Z p ⊗ L is a bijection between the lattices of Q l (with respect to the p-adic valuation on Q * ) and the lattices of Q l p . Hence, any lattice of Q l p may be expressed as the Z p -span of l linearly independent elements of Q l . In particular, there are only countably many lattices of Q l p and so the vector space part map M → V M is countable-to-one. Since any countable-to-one Borel function between standard Borel spaces admits a Borel section, there clearly exists a Borel bijection f : Gr k Q n p × L(Q n−k p ) → M(n, p, k) satisfying V f (V,L) = V for all V, L. It will be useful to work with a particular such function.
4.5. Definition. If V ∈ Gr k Q n p , let V c be the unique complementary subspace of V spanned by basis vectors e j 1 , . . . , e j n−k with the following properties: (a) The Z p -span of (V ∩ Z n p ) ∪ (Z p e j 1 ⊕ · · · ⊕ Z p e j n−k ) is all of Z n p , and (b) j i is the lexicographically greatest sequence satisfying (a).
We call V c the canonical complementary subspace of V.
We now discuss how to find and identify such a sequence j i ; in particular we will show that V c exists. The key is that we can write V as the column space of a n × k matrix A satisfying: • Each row of the k × k identity matrix appears as a row of A (call these the pivot rows), and • Every entry of A is in Z p .
(To obtain such a matrix, begin with an arbitrary matrix whose column space is V. Rescale the first column so that all entries are p-adic integers and at least one entry is 1. Use this 1 to zero out the other entries in its row. Repeat for the second column, etc.) It is easily seen that the sequence j i of indices of the non-pivot rows of A satisfies (a). Our requirement that j i is lex-greatest amounts to the more natural assertion that the sequence of indices of the pivot rows of A is lex-least.
For example, we have already seen in (2.2) that any V ∈ (K p t )V 0 can be written as the column space of a matrix of the form I k v , where the entries of v are in Z p . It follows that V has the canonical complementary subspace V c = Q p e k+1 ⊕ · · · ⊕ Q p e n . Now, for a lattice L ≤ Q n−k p , let L[V] be the isomorphic copy of L inside V c induced by following the obvious map (that is, the linear map defined by e i → e j i ). We define the adjoining of V and L by It is clear from our construction that adjoining defines a Borel bijection which we shall use to parametrize the elements of M(n, p, k).
Relations on k-Grassmann space. Our approach to Lemma 4.1 will be to investigate the "canonical" copy of Gr k Q n p in M(n, p, k). Letting L 0 denote the standard lattice Z n−k p of Q n−k p , we put Y 0 := {(V, L 0 ) : V ∈ Gr k Q n p }. We first give the following characterization. 4.6. Proposition. Y 0 is precisely the orbit (SL n Z p )M 0 , where M 0 is the module Proof. We need only show that for any g ∈ SL n (Z p ), we have g(V 0 , L 0 ) ∈ Y 0 . Then, since SL n (Z p ) acts transitively on Gr k Q n p , it follows that Y 0 is precisely the orbit (SL n Z p )(V 0 , L 0 ). Suppose first that g(V 0 , L 0 ) is of the form (V 0 , L), in other words suppose that gV 0 = V 0 . In this case, g acts on the quotient Q n p /V 0 (in the basis represented by e k+1 , . . . , e n ) via its (n − k) × (n − k) lower right-hand corner g c . It follows easily that g(V 0 , L 0 ) = (V 0 , g c L 0 ), and since the entries of g c lie in Z p , we clearly have is arbitrary. It suffices to show there exists g 1 ∈ SL n (Z p ) such that g 1 (V 0 , L) = (V, L), for then, g −1 1 g(V 0 , L 0 ) = (V 0 , L) and we are in the previous case. Permuting the standard basis if necessary, we can suppose that V = col I k v , where the entries of v are in Z p . It follows easily that g 1 := I k 0 v I n−k satisfies our requirements.

4.7.
Remark. While Y 0 is not invariant for the action of GL n (Q), the last proposition shows in particular that Y 0 is invariant for the action of the subgroup SL n (Z (p) ). However, it is not difficult to see that the restriction of the orbit equivalence relation induced by the action of GL n (Q) on M(n, p, k) to Y 0 is not induced by the action of any subgroup of GL n (Q).
4.8. Proposition. Y 0 is a complete Borel section for the orbit equivalence relation induced by the action of GL n (Q) on M(n, p, k).
Here, a Borel subset A ⊂ X is said to be a complete Borel section for the equivalence relation E on X iff A meets every E-class. A countable Borel equivalence relation is bireducible with its restriction to any complete Borel section. Hence, using the obvious bijection of Y 0 with Gr k Q n p , we obtain that the orbit equivalence relation induced by the action of GL n (Q) on M(n, p, k) is bireducible with the following equivalence relation on Gr k Q n p : Proof of Proposition 4.8. We must prove that for every (V, L) ∈ M(n, p, k), there exists g ∈ GL n (Q) and V ′ such that g(V, L) = (V ′ , L 0 ). (Of course, V ′ will be gV.) Permuting the standard basis if necessary, we may suppose that V = col I k v , where the entries of v are in Z p . Recall that L has a rational basis over Z p , and so there exists h ∈ GL n−k (Q) such that hL = L 0 . Now, choose a rational matrix j so that the entries of j + hv are in Z p , and let g = I k 0 j h . Then both V and gV = col I k j + hv have the canonical complementary subspace V 1 = Q p e k+1 ⊕ · · · ⊕ Q p e n . One now easily computes that g(V, L) = (gV, hL) = (gV, L 0 ), as desired.
Proof of Lemma 4.1. To see that E k SL n Z ⊂ E k ∼ = , suppose that g ∈ SL n (Z) and gV = V ′ . Then by Proposition 4.6, we have g(V, L 0 ) = (V ′ , L 0 ) and so V E k ∼ = V ′ . To see that E k ∼ = ⊂ E k GL n Q , notice that if g ∈ GL n (Q) and g(V, L 0 ) = (V ′ , L 0 ), then gV = V ′ . We have already remarked that the bireducibility of E k ∼ = and ∼ = k n,p follows from Proposition 4.8.

THE PROOFS OF THE MAIN THEOREMS
5.1. Theorem (Theorem B, part 1). If n ≥ 3 and k ≤ n − 2, then ∼ = k n,p ≤ B ∼ k n,p .
By [Tho1,Theorem 4.3], the quasi-isomorphism relation ∼ k n,p is bireducible with E k GL n Q . Using this together with Lemma 4.1, we reduce Theorem 5.1 to the following statement.

Theorem. If n ≥ 3 and k
It is worth remarking that this result gives new examples of countable Borel equivalence relations E ⊂ F such that E ≤ B F (see [Ada]). Before the proof, we introduce a key invariant on the space L(Q l p ) of lattices of Q l p .

Definition.
For a lattice L ∈ L(Q l p ), let A be any l × l matrix over Q p whose columns form a Z p -basis for L. The type of L, denoted tp(L), is the reduction modulo l of ν p (det A). (Here, ν p denotes the p-adic valuation on Q * p .) It is easily checked that the type is independent of the choice of the matrix A. Moreover: In particular, the type of a lattice L depends only on its class Λ = {aL : a ∈ Q p }.

5.5.
Remark. In the case when l = 2, there is a natural graph structure on the set of lattice classes. Join Λ and Λ ′ by an edge iff there are L ∈ Λ and L ′ ∈ Λ ′ such that L ′ is a maximal proper sublattice of L (or vice versa). The resulting graph is the (p + 1)-regular tree and the types correspond to the colors in a 2-coloring of the tree. See the cover of the most recent printing of [Ser] for a picture in the case that l = p = 2.
V 1 = Q p e k+1 ⊕ · · · ⊕ Q p e n . We shall choose the matrix g so that it acts "nontrivially" on V 1 and this will contradict that x E k ∼ = gx. Turning to the details, let g = diag(1, . . . , 1, p), where diag(d 1 , . . . , d n ) denotes the diagonal matrix with a ii = d i . Using equation (2.2) one easily checks that gZ 0 ⊂ Z 0 , and since X 0 = Z 0 modulo a null set, we have that gX 0 is almost contained in X 0 . Together with Lemma 3.7, this implies that we can choose x ∈ Fr(PGL n (Q) Gr k Q n p ) in such a way that x, gx ∈ X 0 ∩ Z 0 . Then x(E k ∼ = )gx, and so Definition 4.9 gives h ∈ GL n (Q) such that (5.6) h(x, L 0 ) = (gx, L 0 ). Now, hx = gx and since we have chosen x so that it is not fixed by any element of PGL n (Q)\{1}, there exists a ∈ Q * p such that h = ag. Since x, gx ∈ Z 0 , each has the canonical complementary subspace V 1 = Q p e k+1 ⊕ · · · ⊕ Q p e n . Since h is diagonal, it clearly acts on V 1 via its (n − k) × (n − k) lower right-hand corner h c . It follows that h(x, L 0 ) = (hx, h c L 0 ), which (together with (5.6)) implies that h c stabilizes L 0 . But it is readily seen that ν p (det h c ) ≡ 1 mod (n − k), so Proposition 5.4 implies that h c does not stabilize L 0 , a contradiction.
We next attack the hardest case of Theorem A. 5.7. Theorem (Theorem A, case 2). Suppose that n ≥ 4 and 2 ≤ k < n/2. Then ∼ = k n,p is Borel incomparable with ∼ = n−k n,p .
This is a consequence of the following slightly stronger result.
Recall from Section 4 that ∼ = l n,p is Borel bireducible with the orbit equivalence relation induced by the action of GL n (Q) on M(n, p, l). In view of the freeness aspects of the proof of Theorem 3.6, one might expect that it would be necessary to work with the action of PGL n (Q). However, PGL n (Q) does not act on M(n, p, l)! Instead, we must work with the action of PGL n (Q) on the space M * (n, p, l) of equivalence classes [M] = {aM : a ∈ Q * p } for M ∈ M(n, p, l). (This is indeed a standard Borel space, since the equivalence relation on M(n, p, k) with equivalence classes [M] is smooth.) Let ( ∼ = l n,p ) * denote the orbit equivalence relation induced by the action of GL n (Q) on M * (n, p, l). 5.9. Proposition. The equivalence relation ( ∼ = l n,p ) * is Borel bireducible with ∼ = l n,p .
Proof. Since the completion map Λ witnesses that ∼ = l n,p is Borel bireducible with the orbit equivalence relation E induced by the action of GL n (Q) on M(n, p, l), it is enough to check that ( ∼ = l The proof now concludes in a manner similar to that of Theorem 5.2. Roughly speaking, we shall choose generic x, gx ∈ X 0 so as to guarantee that f (gx) = g −T f (x). We shall also select g ∈ GL n (Q) so that g −T fails to preserve lattice types on f (X 0 ), contradicting the last claim.
More specifically, let g = diag(1/p, 1, . . . , 1), where diag(d 1 , . . . , d n ) denotes the diagonal matrix with a ii = d i . Arguing as in the proof of Theorem 5.1, we can choose x ∈ X 0 such that gx ∈ X 0 and x ⊥ ∈ Fr(PGL n (Q) Gr n−k Q n p ). Now, both x and gx have the canonical complementary subspace V 1 = Q p e k+1 ⊕ · · · ⊕ Q p e n . Since g acts on V 1 by the identity map, we clearly have g(x, L 0 ) = (gx, L 0 ), and so g witnesses that x E k ∼ = gx. It follows that f (x)( ∼ = n−k n,p ) * f (gx), and hence there exists h ∈ GL n (Q) such that h f (x) = f (gx). Then hf (x) =f (gx), so now The keen-eyed reader will have noticed that Theorem A is as yet incomplete. 5.13. Theorem (Theorem A, case 3). If n ≥ 3 then E 1 ∼ = ≤ B E n−1 ∼ = .
Proof. By Theorem 4.4 of [Tho1], for groups A, B ∈ R(n, p, n−1) we have that A is quasiisomorphic to B iff A is isomorphic to B. In particular, ∼ = n−1 n,p is Borel bireducible with ∼ n−1 n,p , and it follows that E n−1 ∼ = is Borel bireducible with E n−1 GL n Q . Again using the orthogonal complement map, E n−1 GL n Q is Borel bireducible with E 1 GL n Q , and so we have established that the right-hand side E n−1 ∼ = is Borel bireducible with E 1 GL n Q . Hence, the result follows from Theorem B, part 1.