THE CLASSIFICATION OF TORSION-FREE ABELIAN GROUPS OF FINITE RANK UP TO ISOMORPHISM AND UP T

THE CLASSIFICATION OF TORSION-FREE ABELIAN GROUPS OF FINITE RANK UP TO ISOMORPHISM AND UP TO QUASI-ISOMORPHISMSAMUEL COSKEYA BSTRACT.We prove that the isomorphism and quasi-isomorphism relations on the p-local torsion-freeabelian groups offixedfinite rank n are incomparable with respect to Borel reducibility.1.I NTRODUCTIONThis paper extends some recent work(contributors include Hjorth,Kechris,Adams and Thomas) on the complexity of the classification problem for torsion-free abelian groups offinite rank.The clas-sification problem for torsion-free abelian groups of rank1was solved by Baer in1937.The rank2 groups resisted satisfactory classification for sixty years,after which Hjorth used the theory of count-able Borel equivalence relations to prove in[Hjo]that the problem for rank2groups is genuinely more complex than that for ing the ideas of Adams-Kechris[AK],Thomas later proved in[Tho2]that the complexity of the classification problem for torsion-free abelian groups of rank n strictly increases in complexity with n.Thomas also considered the quasi-isomorphism relation on torsion-free abelian groups.Here,A and B are said to be quasi-isomorphic iff A is commensurable with an isomorphic copy of B.As an interme-diate result,Thomas initially proved in[Tho2]that the classification problem for torsion-free abelian groups of rank n up to quasi-isomorphism strictly increases in complexity with n.This left open the question of which of the two classification problems for torsion-free abelian groups offixed rank n is more complex:that up to isomorphism or that up to quasi-isomorphism.In this paper,we prove that if n≥3then the two problems have incomparable complexities.Our result,as well as the others mentioned above,must be expressed using the terminology of countable Borel equivalence relations.Following Friedman-Stanley[FS]and Hjorth-Kechris[HK],in many cases we may regard a“classification problem”as an equivalence relation on a standard Borel space X.1For instance,consider the torsion-free abelian groups offixed rank n.It is easily seen that any such group can be presented as an element of the standard Borel space R(n)consisting of subgroups of Q n that contain a basis for 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 equivalence relation on R(n).We shall denote this equivalence relation∼=n.1A standard Borel space is a separable,completely metrizable space equipped only with itsσ-algebra of Borel sets.1There is a growing theory of countable Borel equivalence relations.Here,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.The equivalence relation E is called countable iff all of its equivalence classes are countable.For example,let us show that∼=n is a countable Borel equivalence relation on R(n).It is easy to verify that subgroups A,B≤Q n are isomorphic if and only if there exists g∈GL n Q such that B=g(A).This immediately implies that∼=n is countable.Since the action of GL n Q on R(n)is Borel(in the sense that for each g∈GL n Q the map x→gx is Borel),it follows that each∼=n is Borel.The equivalence relation∼=n is called the orbit equivalence relation induced by the action of GL n Q on R(n).Here,when a countable groupΓacts in a Borel fashion on a standard Borel space X(which we compactly denoteΓ X),the orbit equivalence relation induced by the Borel actionΓ X is defined byx EΓy⇐⇒x,y lie in the sameΓ-orbitIt is easy to see that ifΓis countable and acts in a Borel fashion,then EΓis Borel.By Feldman and Moore[FM],any countable Borel equivalence relation on a standard Borel space X arises as the orbit equivalence relation induced by the Borel actionΓ X of a countable group.We remark that the groupΓand its action are not canonically determined by EΓ.The case of∼=n is special in the sense that there is a natural group and action inducing it.We now show 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:X→Y such that for all x,x ∈X,x E x ⇐⇒f(x)F f(x )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.Viewing quotient spaces as spaces of possible complete invariants for classification problems,we can also think of≤B as a complexity comparison on Borel spaces as follows:if E≤B F then structurally X/E is a simpler space of invariants than Y/F.Extending this notation in the obvious fashion,we write E∼B F when both E≤B F and F≤B E, we write E<B F when both E≤B F and E∼B F,and we write E⊥B F when both E≤B F and F≤B E.Returning to torsion-free abelian groups,we can now state Hjorth’s and Thomas’s theorems together as∼=1<B ∼=2<B∼=3<B···This was thefirst 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 offinite rank.If p is prime,A is said to be p-local iff it is in-finitely q-divisible for every prime q=p.Let R(n,p)denote the space of p-local groups of rank n and let∼=n,p denote the restriction of∼=n to R(n,p).Thomas’s theorem is that if n≥3and 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].)The proof of this theorem motivates much of what follows.In fact,in our comparison of isomor-phism and quasi-isomorphism,it shall be necessary to restrict to p-local groups.We shall actually restrict to a slightly smaller space,that of p-local torsion-free abelian groups offixed 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 space of p-local torsion-free abelian groups of rank n and of divisible rank k.Let∼=k n,p denote the restrition of∼=n to R(n,p,k).Our first key result is the following:T HEOREM A.Fix n≥3and p prime.Then for k=l,we have that∼=k n,p is Borel incomparable with∼=l n,p. We next consider the quasi-isomorphism relation on the spaces R(n,p,k).Here,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 hasfinite index in both A,B ).In the case k=n−1, the notion of quasi-isomorphism coincides with that of isomorphism.For k<n−1they are distinct. Thomas found the quasi-isomorphism relation easier to work with in[Tho2],for reasons which will be clear in later sections.However,the next theorem shows that the classification of(local)groups up to quasi-isomorphism is not simpler than that up to isomorphism.Let∼k n,p denote the quasi-isomorphism equivalence relation on the space R(n,p,k)of p-local torsion-free abelian groups of rank n and divisible rank k.Then we have:T HEOREM B.If1≤k≤n−2,then∼=k n,p is Borel incomparable with∼k n,p.It follows easily from these two results,together with Theorem4.7of[Tho1]that for n≥3,the isomorphism and quasi-isomorphism relations on on the space of all local(that is,p-local for any p) torsion-free abelian groups of rank n are incomparable as well.C ONJECTURE.For n≥3,the isomorphism and quasi-isomorphism relations on the space R(n)of torsion-free abelian groups of rank n are Borel incomparable.This paper is organized as follows.In the second section,we introduce the notion of an affine map between 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 over the p-adics.We state a lemma(proved in the ap-pendix)characterizing the affine maps between the k-Grassmann spaces.In the third section,we state a cocycle superrigidity theorem of Adrian Ioana,and immediately derive a corollary which roughly states that a homomorphism of SL n Z-orbits between Grassmann spaces is a slight perturbation of an action-preserving map.We use this to prove the auxilliary result that the orbit equivalence relations induced by the action of GL n Q on the Gr k Q n p are Borel incomparable as k varies.In the fourth section, we define the completion of torsion-free abelian groups A≤Q n to Z p-submodules of Q n p.(This method was employed by Kurosh and Malcev in their attempts to classify the torsion-free abelian groups of finite rank.)This will lead us to transfer the domain of discourse from the complicated R(n,p,k)to the space of completed groups and later to the much more reasonable k-Grassmann space.In the last section,we prove Theorems A and B.2.E RGODIC THEORY OF HOMOGENEOUS SPACESIn this section,we define the the notion of ergodicity,which plays an essential role in the theory of countable Borel equivalence relations.Ergodicity and Borel reductions.We now define the notion of ergodicity of a measure-preserving action.LetΓbe a countable group acting in a Borel fashion on the standard Borel space X.If X carries a probability measureµ,then we writeΓ (X,µ)to indicate thatΓacts on X in aµ-preserving fashion.As before,we let EΓdenote the orbit equivalence relation on X induced by the action ofΓ. We say that the actionΓ (X,µ)is ergodic iff everyΓ-invariant subset of X hasµ-measure0or1. Clearly,this property depends only on the orbit equivalence relation EΓand the measureµ.We shall use the characterization thatΓ (X,µ)is ergodic iff for everyΓ-invariant function f:X→Y into a standard Borel space Y,there exists aµ-conull A⊂X such that f|A is a constant function.This notion has a generalization that will arise in our arguments.First,we introduce the notion of homomorphism of orbits,a weakening of Borel reduction.Let E,F be equivalence relations on standard Borel spaces X,Y.A homomorphism of orbits f:E→F is any map satisfying justx E y=⇒f(x)F f(y)A homomorphism of orbits is similar to a Borel reduction but with“collapsing”allowed.This cor-responds to using Y/F as a space of incomplete invariants for the E-classification problem on X.Acountable-to-one homomorphism of orbits E→F is called a weak reduction from E to F.We write E≤w F if there exists a(Borel)weak reduction E→F.Notice thatΓ (X,µ)is ergodic iff every Borel homomorphism of orbits EΓ→Id Y is constant on a conull set.(Here,Id Y represents the equality relation on the standard Borel space Y.)More generally, if F is a Borel equivalence relation on the standard Borel space Y,then we sayΓ (X,µ)is F-ergodic iff for every Borel homomorphism of orbits f:EΓ→F,there exists aµ-conull subset A⊂X such that f(A)is contained in a single F-class.It is worth noting that:E is F-ergodic=⇒E≤w F=⇒E≤B FErgodic components.IfΓ (X,µ)is ergodic andΛ≤Γis a subgroup offinite index,then there exists a partition X=Z1 ··· Z N of X intoΛ-invariant cells,such thatµZ i>0andΛ (Z i,µi)is ergodic.Here,µi denotes the(normalized)measure induced on Z i byµ.The cells Z i(together with the actionΛ Z i)are called ergodic components forΛ;they are determined uniquely up to null sets by Λand the actionΓ (X,µ).Homogeneous spaces.Let K be a compact,second countable group.If K acts transitively on the standard Borel K-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 measure(the Haar measure),it follows that X does as well.Now suppose that Γ≤K is a countable dense subgroup.Then the actionΓ K/L is uniquely ergodic,meaning that the Haar measure isΓ-ergodic and moreover it is the unique probability measure on K/L that is preserved byΓ.Now,suppose thatΛ≤Γis a subgroup offinite index.Then the ergodic components for the action ofΛare precisely the orbits of¯Λ(the closure in K)on K/L and each ergodic component is again a homogeneous space for the compact group K0=¯Λ.The action ofΛon the particular component K0L is isomorphic to that on K0/L0,where L0=L∩K0.IfΛ Γis a normal subgroup offinite index,then Γacts as a transitive permutation group on the¯Λ-orbits,i.e.,on the ergodic components forΛ. Finally we define the natural morphisms between homogeneous spaces.For i=0,1let K i be compact and L i a closed subgroup.Then a function f:K0/L0→K1/L1is said to be affine iff there exists an isomorphismφ:K0→K1such that f(kx)=φ(k)f(x)for all x∈K0/L0and k∈K0.In this case,there exists t∈K1with the property that:•φ(L0)=tL1t−1,and•f(kL0)=φ(k)tL1for k∈K0.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 ≤Aut Q n p acts transitively on Gr k Q n p (see [Tho3],Proposition 6.1)we can view Gr k Q n p as a homogeneousSL n Z p -space.Accordingly,it carries a corresponding Haar 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 m th principal congruence subgroup Γm SL n Z is defined by:(2.1)Γm =ker[SL n Z →SL n (Z /m Z )]where the map on the right-hand side is the canonical surjection.It is easily seen that Γm =K m ,where (2.2)K m =ker[SL n Z p →SL n (Z p /m Z p )]Hence,the ergodic components of Gr k Q n p corresponding to the action of Γm are precisely the K m -orbits.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 column space of a matrix a v,where a is congruent to the k ×k identity matrix I k modulo p t ,and v is congruent to 0modulo p t .Since a is clearly invertible,one can use column operations to suppose that a =I k .So wehave:(2.3)K p t .V 0= col I k v:p t |v where p t |v means that for each entry x of v ,we have that x/p t lies in Z p .We close this section with the following lemma,which identifies all affine maps between ergodic components of Grassmann space.The proof is given in the Appendix.2.4.L EMMA .Let n ≥3and X 0=K 0/L 0and X 1=K 1/L 1be ergodic components for Gr k Q n p ,Gr l Q n p ,respectively.Let f :X 0→X 1be an affine map.Then l =k or l =n −k ,and:•In the case l =k =n −k ,there exists h ∈GL n Q such that f satisfies f (x )=hx almost everywhere;•In the case l =n −k =k ,there exists h ∈GL n Q such that f satisfies f (x )=hx ⊥almost everywhere,where x ⊥denotes the orthogonal complement of x .3.A SUPERRIGIDITY THEOREMIn this section,we introduce the following recent cocycle superrigidity theorem of Adrian Ioana(see [Ioa]).We then derive a corollary which will be used in the proofs of our main theorems.Familiarity with Borel cocycles is only needed to understand the statement of Ioana’s theorem,and the proof of the corollary.We remark that it would also be possible to use Zimmer’s cocycle superrigidity theorem in our arguments.To do so would require more techniques,such as induced cocycles andfinite ex-tensions.In this context,Ioana’s theorem is much more powerful and hence our arguments shall not require such maneuvres.3.1.D EFINITION.For i∈N letΓi (X i,µi)andρi:X i+1→X i be a factor map(i.e.,aΓ-invariant measure-preserving map).Then the corresponding inverse limit is aΓ-space(X,µ)together with factor mapsπi:X→X i satisfyingπi=ρi◦πi+1and the usual universal property associated with inverse limits.In case all X i arefinite sets,(X,µ)is called a profiniteΓ-space.3.2.D EFINITION.The action of a countable groupΓon the standard Borel space X is free if it isfixed-point free,meaning that if1=g∈Γthen gx=x for all x∈X.3.3.I OANA’S T HEOREM.Suppose thatΓbe a countable Kazhdan group,and let(X,µ)be a free,ergodic profiniteΓ-space.Letπi:X→X i be the factor maps from Definition3.1.Suppose thatα:Γ X→Λis a cocycle into an arbitrary countable groupΛ.Then there exists i∈N such thatαis cohomologous to a cocycle Γ X i→Λ.More precisely,there exists i∈N and a cocycleαi:Γ X i→Λsuch thatαis cohomologous to the cocycle α defined byα (g,x)=αi(g,πi(x)).The corollary below addresses the case thatαis a cocycle corresponding to a Borel homomorphism of orbits f into an orbit equivalence relation induced by a free action ofΛ.We shall need the following definition.Suppose thatΓ X andΛ Y are Borel actions of countable groups on standard Borel spaces.Then a homomorphism of permutation groups(φ,f):Γ X−→Λ Y is a pair(φ,f),where φ:Γ→Λis a group homomorphism and f:X→Y is a Borel map satisfying f(gx)=φ(g)f(x)for all g∈Γand x∈X.3.4.C OROLLARY.Suppose thatΓbe a countable Kazhdan group,and let(X,µ)be a free,ergodic profinite Γ-space.LetΛbe a countable group andΛ Y a free action.Suppose that f:EΓ−→EΛis a Borel homomorphism of orbits.Then there exists an ergodic componentΓ0 X0forΓ X and a permutation group homomorphism(φ,f ):Γ0 X0−→Λ Y such that for all x∈X0,f(x)is EΛ-equivalent to f (x).P ROOF.Letα:Γ X→Λbe the cocycle corresponding to f.By the theorem,there exists afinite factor(X ,µ )of(X,µ)(say the projection map isπ),and a map b:X→Λsuch that the adjusted cocycleα (g,x)=b(gx)α(g,x)b(x)−1depends only on g,π(x).LetΓ0be the stabilizer of any point s∈X such thatµ {s}>0,so that S:=π−1{s}isΓ0-invariant andµ(S)>0.Then S contains an ergodic component X0for the action ofΓ0.Sinceπis constant on X0,the restriction ofα to the action Γ0 X0is independent of x∈X0.It follows thatα defines a homomorphismφ:Γ0→Λ.Letting f =bf,it is easily seen that(φ,f )satisfies our requirements.(Of course,since the cocycle identity holds almost everywhere,one may need to delete a null set of X0.) Wefirst apply this corollary to establish the following Borel incomparability.It will serve as an auxilliary theorem in the next section,but it is also interesting in its own right.3.5.N OTATION.Fix n∈N and p prime.For the rest of this section,E k SLn Z and E k GLn Qdenote the orbitequivalence relations induced on Gr k Q n p by SL n Z and GL n Q,respectively.3.6.T HEOREM.Suppose that k,l≤n and k=l,n−l.Then E k GLn Q is Borel incomparable with E l GLn Q.We shall see in the next section that this theorem implies that the quasi-isomorphism relations on R(n,p,k),R(n,p,l)are Borel incomparable.Theorem3.6follows immediately from the following slightly stronger statement.The proof offers a preview of the arguments found in the last section.3.7.T HEOREM.Suppose that k,l≤n and k=l,n−l.Then E k SLn Z(together with the Haar measure on Gr k Q n p)is E l∼-ergodic.P ROOF.Let f:E k SLn Z→E l∼be a Borel homomorphism of orbits.Thefirst step is to apply Corollary3.4.While it is not literally true that GL n Q Gr l Q n p is free,in our context it suffices to establish thatf maps a(Haar)conull set into the free part of the action of P GL n Q on Gr l Q n p.Here,if the countable groupΓacts on a standard Borel space X,thenF r(Γ X)={x∈X|1=g∈Γ=⇒gx=x}denotes the free part of the action ofΓon X.The next lemma establishes this precisely.The statement is adapted from that of Lemma5.1of[Tho1],and the proof can be adapted as well.We remark that Zimmer’s superrigidity theorem is used in the proof.3.8.L EMMA(Thomas).Suppose that f:E k SLn Z →E lGL n Qis a Borel homomorphism of orbits.Then either ftakes values in a single GL n Q-orbit,or there is a conull M⊂Gr k Q n p such that f(M)⊂F r(P GL n Q Gr l Q n p). It also follows from the lemma that F r(SL n Z Gr k Q n p)is itself conull.Hence,we may apply Corollary3.4to suppose that there exists an ergodic componentΓ0 X0of SL n Z Gr k Q n p and ahomomorphismφ:Γ0→GL n Q such that(φ,f):Γ0 X0−→GL n Q Gr l Q n p is a homomorphism of permutation groups.The next step is to show that without loss of generality,im(f)is contained in an ergodic component of SL n Z Gr l Q n p.For this,we appeal to Margulis’s superrigidity theorems.3.9.L EMMA.IfΓ0≤SL n Z is afinite index subgroup andφ:Γ0→GL n Q is a homomorphism,then there exists afinite index subgroupΛ≤Γ0such thatφ(Λ)⊂SL n Z.To simplify the proof,let us suppose that n is odd and so SL n R is a simple group.P ROOF.SinceΓ0is Kazhdan,Γ 0:=[Γ0,Γ0]is afinite index subgroup ofΓ0(see[Lub],Corollary1.29). Now since GL n Q/SL n Q∼=Q×is abelian,we have thatφ(Γ 0)≤[GL n Q,GL n Q]≤SL n Q(In fact,the latter≤is an equality.)ReplacingΓ0byΓ 0if necessary,we may suppose without loss of generality that imφ⊂SL n Q.We can also suppose without loss of generality that imφis infinite.By Theorem VIII.3.10of[Mar], the Zariski closure H ofφ(Γ0)in SL n R is semisimple.Letπi:H→H i denote the projections of H onto its simple factors.By the Mostow-Margulis superrigidity theorem(see Theorem5.1of[Zim]), eachφi:=πi◦φextends to a homomorphismΦi:SL n R→H i.Since SL n R is almost simple,there is exactly oneΦi with infinite image.It follows that the corresponding factor H i is actually all of SL n R. Hence we have thatφextends to an automorphismΦof SL n R.Clearly,φ(Γ0)is again a lattice.By IX.4.14of[Mar],sinceφ(Γ0)⊂SL n Q it is commensurable with SL n Z.In particular,there exists afinite index subgroupΛ≤Γ0such thatφ(Λ)⊂SL n Z. Hence,replacingΓ0 X0with a smaller ergodic component,we can suppose thatΓ1:=im(φ)⊂SL n Z.Now,sinceΓ0 X0is ergodic,there exists an ergodic component X1for the action ofΓ1on Gr l Q n p such that im(f)⊂X1.Recall that X i∼=K i/L i asΓi-spaces,where K i is the closure ofΓi in SL n Z p and L i≤K i are closed subgroups.Claim.φextends to an isomorphism K0→K1.We can suppose thatφ:Γ0→Γ1is injective.(If not,then by Margulis’s theorem on normal sub-groups the kernel ofφhasfinite index inΓ0.Hence,φhasfinite image and so passing to an ergodic subcomponent,we would be able to suppose thatφ=1.This implies that f isΓ0-invariant and since Γ0 X0is ergodic,f is almost constant.)By Lemma6.1of the appendix,we can suppose thatφis con-jugation by an element of GL n Q and soφextends to an automorphism of GL n Q p.Sinceφ(K0)is a com-pact group containingΓ1,we have thatφ(K0)⊃K1.The same argument shows thatφ−1(K1)⊃K0,and soφ(K0)=K1.This completes the claim.(It is possible to give a more self-contained argument adapted from Proposition7.2of[Fur].)Now,define a mapβ:K0→K1/L1byβ(k)=φ(k)−1f(kL0)Notice thatβisΓ0-invariant:forγ∈Γ0,we haveβ(γk)=φ(γk)−1f(γkL0)=φ(k)−1f(kL0)=β(k). Hence by ergodicity ofΓ0 X0,there exists t∈K1such thatβ(k)=tL1for(Haar)almost every k∈K0.In other words,we can suppose that f(kL0)=φ(k)tL1is an affine map.Since we have assumed that k=l,n−l,this contradicts Lemma2.4.≤w P ROOF OF T HEOREM3.6.Suppose that f:E k∼≤B E l∼.Then clearly f is a weak reduction E k SLn ZE l∼.By Theorem3.7,f carries a conull set M into a countable set.Since f is a weak reduction,it follows that M is countable.Since the Haar measure or Gr k Q n p is nonatomic,this is a contradiction.4.C OMPLETION OF TORSION-FREE ABELIAN GROUPSThe completion map.Recall that A∈R(n)is p-local iff it is q-divisible for all q=p group.In this section,we summarize some results found in[Fuc]regarding the following completion of A to a Z p-submodule of Q n ly,define(4.1)Λp(A):=A⊗Z pThat is,considering A as a subset of Q n p,Λp(A)is just the set Z p-linear combinations of elements of A. We shall omit the subscript p when it is clear from context.On R(n,p),Λ=Λp 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,by93.1 and93.5of[Fuc],Λis a bijection of R(n,p)with M(n,p).Suppose now that M∈M(n,p).Then by93.3of[Fuc],M can be decomposed as a direct sum (4.2)M=V M⊕Lwhere 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 well-defined from M.There are many possible complementary submodules L,but in any case we have rank L=n−dim V.By exercise93.1of[Fuc],the dimension of VΛ(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 thatΛis a bijection of R(n,p,k)with M(n,p,k).4.3.D EFINITION .These facts allow us to transfer the domain of the isomorphism relation from the space R (n,p )of p -local subgroups of Q n to the space M (n,p )of Z p -submodules of Q n p .In other words,if M 0,M 1∈M (n,p )and M i =Λ(A i )then we shall write M 0∼=n,p M 1iff A 0∼=n,p A 1.The two uses of ∼=n,p are Borel equivalent ,meaning that there is a Borel bijection between their domains which identifies equivalence classes.Since Λpreserves the action of GL n Q p ,each is induced by the relevant action of GL n Q .Similarly,we write M 0∼n,p M 1iff A 0∼n,p A 1.By Theorem 4.8of [Tho2],the two uses of the ∼n,psymbol are Borel equivalent.Of course,we also write ∼=k n,p ,∼k n,p to denote the restrictions of ∼=n,p ,∼n,pto M (n,p,k ).The adjoining map.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 =0and V ⊕W =Q n p ,then M can always be written (uniquely)as V ⊕L where L ≤W .Hence the set {M ∈M (n,p,k ):V M =V }is in bijection with the lattices of W ,i.e.,the free Z p -submodules of W that contain a basis for W .In this section,we will choose a canonical complementary subspace V c ,and list some properties.Still fixing V ∈Gr k Q n p ,let j 1...j n −k be a subsequence of 1,...,n satisfying:(4.4)The Z p span of (V ∩Z n p )∪(Z p e i 1⊕···⊕Z p e i k )is all of Z n p .We will show that such a sequence exists in the next paragraph.To make it so that the j i are determined by V ,we arbitrarily require that j i is the lexicographically greatest sequence satisfying (4.4).DefineV c =Q p e j 1⊕···⊕Q p e j n −k .Then V c is complementary to V and it is canonically isomorphic to Q n −k p.We now show how to find and identify the sequence j i .It is possible to 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 wih an arbitrary matrix whose column space is V .Rescale the first column so that all entries are integers and at least one entry is e this 1to 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 (4.4).For example,we have already seen in (2.3)that any V ∈K p t .V 0can 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 the canonical complementary subspace is 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 the abuse of notation e i →e j i ).Now we define theadjoining of V and L by(4.5)(V,L):=V⊕L[V]Let L(Q n−kp )denote the set of lattices of Q n−kp.The following proposition summarizes our work thusfar,and allows us to identify M(n,p,k)with a product of Gr k Q n p and L(Q n−kp).4.6.P ROPOSITION.The adjoining map(·,·)yields a bijection Gr k Q n p×L(Q n−kp)→M(n,p,k).This means that any M∈M(n,p,k)can be uniquely written M=(V M,L M),where V M is the vectorspace part of M.We shall call L M the lattice part of M.P ROOF.For injectivity,suppose that(V,L)=(V ,L ).Clearly,V=V so that L[V]∩V c=L [V]∩V c, and hence L=L .For surjectivity,let M∈M(n,p,k)be arbitrary.Let V c be as above and letι:Q n−kp→V c be the transfer of bases isomorphism.Then L:=ι−1(V c∩M)satisfies M=(V,L). Lattices and types.Any lattice L∈L(Q l p)may be expressed as the Z p-span of l linearly independant vectors in Q l p.By the discussion of completion in[Ser],section II.1.1,these vectors may be chosen in Q n.In particular,there are countably many lattices.If L is the span of the vectors x1,...,x l∈Q l p,let A be the matrix whose columns are the x i.The type of L is defined as the reduction modulo l ofνp(det A),whereνp denotes the p-adic valuation on Q∗p.(It is easily checked that this is independant of the choice of the x i.)It is immediate that for any s∈GL l Q p,we have tp(sL)≡νp(det s)+tp(L),modulo l.It follows that the group of type-preserving maps is precisely the kernel of the composition(4.7)GL l Q p det−→Q∗pνp−→Z−→Z/l ZUsing this,we can establish:4.8.P ROPOSITION.If g∈SL n Z p and g(V,L)=(gV,L ),then tp(L)=tp(L ).P ROOF.Let V0=Z p e1⊕···⊕Z p e k denote the standard subspace of Q n p.Wefirst treat the case that V,gV=V0.In this case,gV0=V0and it follows that g acts as a linear transformation on V c0= Q p e k+1⊕···⊕Q p e n via its n−k×n−k lower right-hand corner,denote it g c.Clearly,g c∈GL n−k Z p and so it preserves types.Hence,tp(L )=tp(g c L)=tp(L).Now,we treat the case that V=V0but gV is arbitrary(the general case follows by an easy diagram). Let g(V0,L)=(gV0,L ).It suffices to show there exists g1∈SL n Z p such that g1(V0,L )=(gV0,L ). For then,g−1g1(V0,L )=(V0,L)and we are in the previous case.Permuting the standard basis if。