aa r X i v : . [ m a t h . L O ] F e b TORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE
GIANLUCA PAOLINI AND SAHARON SHELAH
Abstract.
We prove that the Borel space of torsion-free Abelian groups withdomain ω is Borel complete, i.e., the isomorphism relation on this Borel spaceis as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the sem-inal paper on Borel reducibility of Friedman and Stanley from 1989. Introduction
Since the seminal paper of Friedman and Stanley on Borel complexity [3], descrip-tive set theory has proved itself to be a decisive tool in the analysis of complexityproblems for classes of countable structures. A canonical example of this phenom-enon is the famous result of Thomas from [12] which shows that the complexity ofthe isomorphism relation for torsion-free abelian groups of rank 1 n < ω (denotedas ∼ = n ) is strictly increasing with n , thus, on one hand, finally providing a satisfac-tory reason for the difficulties found by many eminent mathematicians in findingsystems of invariants for torsion-free abelian groups of rank 2 n < ω which wereas simple as the one provided by Baer for n = 1 (see [1]), and, on the other hand,showing that for no 1 n < ω the relation ∼ = n is universal among countable Borelequivalence relations. As a matter of facts, abelian group theory has been one of themost important fields of mathematics from which taking inspiration for forging thegeneral theory of Borel complexity as well as for finding some of the most strikingapplications thereof. The present paper continues this tradition solving one of themost important problems in the area, a problem open since the above mentionedpaper of Friedman and Stanley from 1989. In technical terms, we prove that thespace of countable torsion-free abelian groups with domain ω is Borel complete .As we see will in detail below, saying that a class of countable structures is Borelcomplete means that the isomorphism relation on this class is as complicated aspossible, as an isomorphism relation. The Borel completeness of countable abeliangroup theory is particularly interesting from the perspective of model theory, as thisclass is model theoretically “low”, i.e stable (in the terminology of [10]). In fact, asalready observed in [3], Borel reducibility can be thought of as a weak version of L ω ,ω -interpretability, and for other classes of countable structures such as groupsor fields much stronger results than Borel completeness exist, as in such cases we canfirst-order interpret graph theory, but such classes are unstable, while abelian grouptheory is stable. Reference [8] starts a systematic study of the relations betweenBorel reducibility and classification theory in the context of ℵ -stable theories. Date : February 25, 2021.No. 1205 on Shelah’s publication list. Research of both authors partially supported by NSFgrant no: DMS 1833363. Research of the first author partially supported by project PRIN 2017“Mathematical Logic: models, sets, computability”, prot. 2017NWTM8R. Research of the secondauthor partially supported by Israel Science Foundation (ISF) grant no: 1838/19.
Coming back to us, we now introduce the notions from descriptive set theorywhich are necessary to understand our results, and we try to make a completehistorical account of the problems which we tackle in this paper. The startingpoint of the descriptive set theory of countable structures is the following fact:
Fact 1.1.
The set K Lω of structures with domain ω in a given countable language L is endowed with a standard Borel space structure (K Lω , B ) . Every Borel subset ofthis space (K Lω , B ) is naturally endowed with the Borel structure induced by (K Lω , B ) . For example, if take L = { e, · , () − } , and we let K ′ to be one of the following:(a) the set of elements of K Lω which are groups;(b) the set of elements of K Lω which are abelian groups;(c) the set of elements of K Lω which are torsion-free abelian groups;(d) the set of elements of K Lω which are n -nilpotent groups, for some n < ω ;then we have that K ′ is a Borel subset of (K Lω , B ), and so Fact 1.1 applies.Thus, given a class K ′ as in Fact 1.1, we can consider K ′ as a standard Borelspace, and so we can analyze the complexity of certain subsets of this space or ofcertain relations on it (i.e., subsets of K ′ × K ′ with the product Borel space struc-ture). Further, this technology allows us to compare pairs of classes of structuresor, in another direction, pairs of relations defined on pairs of classes of structures. Definition 1.2.
Let X and X be two standard Borel spaces, and let also Y ⊆ X and Y ⊆ X . We say that Y is Borel reducible to Y , denoted as Y R Y , whenthere is a Borel map B : X → X such that for every x ∈ X we have: x ∈ Y ⇔ B ( x ) ∈ Y . Notice that Definition 1.2 covers in particular the case X = K ′ × K ′ for K ′ asin Fact 1.1, and so for example Y could be the isomorphism relation on K ′ . Also,given a Borel space X , we can ask if there are subsets of X which are R -maximawith respect to a fixed family of subsets of an arbitrary Borel space (e.g., Borelsets, analytic sets, co-analytic sets, etc). In particular we can define: Definition 1.3.
Let X be a Borel space and Y ⊆ X . We say that Y is completeanalytic (resp. complete co-analytic) if for every Borel space X and analytic subset(resp. co-analytic subset) Y of X we have that Y R Y . Similarly, we can ask if in a given class of countable structures the isomorphismrelation is as complex as possible among all isomorphism relations arising fromclasses of countable structures. This is made precise by the notion of
Borel com-pleteness , the above mentioned notion which is at the heart of our main theorem.
Definition 1.4.
Let K be a Borel class of structures with domain ω . We say that K is Borel complete (or
Sym( ω ) -complete) if for every Borel class K of structureswith domain ω there is a Borel map B : K → K such that for every A, B ∈ K : A ∼ = B ⇔ F ( A ) ∼ = F ( B ) . The following fact will be relevant for our subsequent historical account.
Fact 1.5 ([3]) . Let K be a Borel class of structures with domain ω . If K is Borelcomplete, then its isomorphism relation is a complete analytic subset of K × K , butthe converse need not hold, as for example abelian p -groups with domain ω havecomplete analytic isomorphism relation but they are not a Borel complete space. ORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE 3
We now have all the ingredients necessary to be able to understand the problemsthat we solve in this paper and to introduce the state of the art concerning them.But first a useful piece of notation which we will use throughout the paper.
Notation 1.6. (1) We denote by
Graph the class of graphs.(2) We denote by Gp the class of groups.(3) We denote by AB the class of abelian groups.(4) We denote by TFAB the class of torsion-free abelian groups.(5) Given a class K we denote by K ω the set of structures in K with domain ω . Convention 1.7.
To simplify statements, we use the following convention: whenwe say that a class K of countable structures is Borel complete we mean that K ω is Borel complete. Similarly, when we say that a class K of countable groups iscomplete co-analytic we mean that K ω is a complete co-analytic subset of Gp ω .Finally, when we say that the isomorphism relation on a class of countable groupsis analytic, we mean that restriction of the isomorphism relation on K to K ω × K ω is an analytic subset of the Borel space Gp ω × Gp ω (as a product space). In [3], together with the general notions just defined, the authors studied someBorel complexity problems for specific classes of countable structures of interest.Among other things they showed (we mention only the results relevant to us):(i) countable graphs, linear orders and trees are Borel complete;(ii) torsion abelian groups are complete analytic but not
Borel complete;(iii) nilpotent groups of class 2 and exponent p ( p a prime) are Borel complete ;(iv) the isomorphism relation on finite rank torsion-free abelian groups is Borel.In [3] Friedman and Stanley state explicitly:There is, alas, a missing piece to the puzzle, namely our conjecturethat torsion-free abelian groups are complete. [...] We have noteven been able to show that the isomorphism relation on torsion-free abelian groups is complete analytic, nor, in another direction,that the class of all abelian groups is Borel complete. We considerthese problems to be among the most important in the subject.The challenge was taken by several mathematicians. The first to work on thisproblem was Hjorth, which in [6] proved that any Borel isomorphism relationis Borel reducible to the isomorphism relation on countable torsion-free abeliangroups, and that in particular the isomorphism relation on TFAB ω is not Borel,leaving though open the question whether TFAB ω is a Borel complete class, or evenwhether the isomorphism relation on TFAB ω is complete analytic (cf. Fact 1.5).The problem resisted further attempts of the time and the interest moved toanother very interesting problem on torsion-free abelian groups: for 1 n < m < ω ,is the isomorphism relation on torsion-free abelian groups of rank n strictly lesscomplex than the isomorphism relation on torsion-free abelian groups of rank m ?As mentioned above, the isomorphism relation on torsion-free abelian groups offinite rank is Borel while, as just mentioned, the isomorphim relation on countabletorsion-free abelian groups is not, and so the two problems are quite different, butobviously related. Also this problem proved to be very challenging, until Thomasfinally gave a positive solution to the problem, in a series of two fundamental papers As already mentioned in [3], this result is actually a straightforward adaptation of a modeltheoretic construction due to Mekler [9].
GIANLUCA PAOLINI AND SAHARON SHELAH [11, 12]. In [11, 12] Thomas, continuing on work of Hjorth and Kechris [7], alsoproved interesting results on the isomorphism relation on rigid torsion-free abeliangroups of finite rank, where, in their terminology, an abelian group G is said to berigid when the only automorphisms of G are the identity and the map g
7→ − g .This notion of rigidity is relevant also to our main theorem, as we will show thatthe Borel map witnessing our Borel complete construction can be taken to haverange in the rigid abelian torsion-free groups, we will return to this later.The fundamental work of Thomas thus resolved completely the case of torsion-free abelian groups of finite rank, leaving open the problem for countable torsion-free abelian groups of arbitrary rank, i.e. the problem referred to as “among themost important in the subject” in [3]. The problem remained dormant for variousyears (at the best of our knowledge), until Downey and Montalb´an [2] made someimportant progress showing that the isomorphism relation on countable torsion-free abelian groups is complete analytic, a necessary but not sufficient condition forBorel completeness, as recalled in Fact 1.5. This was of course possible evidencethat the isomorphism relation was indeed Borel complete, as conjectured in [3].Despite this advancement, the problem of Borel completeness of countable torsion-free abelian groups resisted for other 12 years, until this day, when we prove: Main Theorem.
The space
TFAB ω is Borel complete, in fact there exists a Borelmap B : Graph ω → TFAB ω such that for every H , H , ∈ Graph ω we have that:(i) H ∼ = H if and only if B ( H ) ∼ = B ( H ) ;(ii) B ( H ) has only trivial automorphisms (i.e. the identity and g
7→ − g ). As already mentioned, clause (ii) of our Main Theorem, shows that the Borelfunction witnessing Borel reducibility of the isomorphism relation on Graph ω tothe isomorphism relation on TFAB ω can be taken to have range in the subset ofrigid abelian groups, in the sense of Hjorth, Kechris and Thomas mentioned above.In fact this might be considered of independent interest and it is the infinite rankcounterpart of the results of Hjorth, Kechris and Thomas relating the isomorphismrelation on rigid and general torsion-free abelian groups of finite rank, e.g. Thomasfamously showed that the isomorphism relation on the rigid TFAB ω of rank n + 1does not Borel reduces to the isomorphism relation on TFAB ω of rank n > ω lead us to the consideration of classificationquestions of co-Hopfian torsion-free abelian groups. We recall that a group G issaid to be co-Hopfian if G does not have proper subgroups H isomorphic to G ,i.e., every injective endomorphism of G is surjective (and thus an automorphism).As well-known (see e.g. [4, Proposition 2.2, pg. 130]), for G ∈ TFAB, G is co-Hopfian iff G is divisible and of finite rank, i.e. G is a finitely dimensional vectorspace over Q , and so clearly the co-Hopfian groups form a Borel subset of TFAB ω .We wonder: what if replace the notion of surjective morphism with a notion of“almost-surjective” morphism which is appropriate for the class TFAB? Does theclassification problem becomes intractable? In particular we might consider: Definition 1.8. (1) We define the collection
Emb of embeddings between ele-ments of TFAB as f : G → H ∈ Emb if and only if H/f [ G ] is torsion.(2) We define the maps Emb on TFAB as those f : G → H ∈ Emb such that f [ G ] is H/f [ G ] torsion and bounded (i.e., there is n ∈ ω such that nf [ G ] = 0 ).(3) We define Emb as those f ∈ Emb of the form g mg for some m ∈ Z \ { } . ORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE 5
These three notions of “almost-surjective” morphism lead to three variations ofthe notion of co-Hopfian group (cf. Definition 2.7) and for them we are able to show:
Theorem 1.9.
For ℓ ∈ { , , } , the set of Emb ℓ -co-Hopfian torsion-free abeliangroups is a complete co-analytic subset of the Borel space space TFAB ω . In a work in preparation we extend the ideas behind Theorem 1.9 to a systematicinvestigation of various classification problems for (co-)Hopfian abelian and nilpo-tent groups from the perspective of descriptive set theory of countable structures.2.
Notations and Preliminaries
For the readers of various backgrounds we try to make the paper self-contained.2.1.
General notationsDefinition 2.1. (1) Given a set X we write Y ⊆ ω X for Y ⊆ X and | Y | < ℵ .(2) Given a set X and ¯ x, ¯ y ∈ X <ω we write ¯ y ⊳ ¯ x to mean that lg(¯ y ) < lg(¯ x ) and ¯ x ↾ lg(¯ y ) = ¯ y , where ¯ x is naturally considered as a function X lg(¯ x ) → X .(3) Given a partial function f : M → M , we denote by dom( f ) and ran( f ) thedomain and the range of f , respectively.(4) For ¯ a ∈ B n we write ¯ x ⊆ B to mean that ran(¯ x ) ⊆ B , where, as usual, ¯ a isconsidered as a function { , ..., n − } → B .(5) Given a sequence ¯ f = ( f i : i ∈ I ) we write f ∈ ¯ f to mean that there exists j ∈ I such that f = f j . GroupsNotation 2.2.
Let G and H be groups.(1) H G means that H is a subgroup of G .(2) We let G + = G \ { e G } , where e G is the neutral element of G .(3) If G is abelian we might denote the neutral element e G simply as G = 0 . Definition 2.3.
Let H G be groups, we say that H is pure in G , denoted by H ∗ G , when if k ∈ H , n < ω and (in additive notation) G | = ng = k , then thereis h ∈ H such that H | = nh = g . Observation 2.4. H ∗ G ∈ TFAB , k ∈ H , < n < ω , G | = ng = k ⇒ g ∈ H . Observation 2.5.
Let G ∈ TFAB and let: G (1 ,p ) = { a ∈ G : a is divisible by p m , for every < m < ω } , then G (1 ,p ) is a pure subgroup of G .Proof. This is well-known, see e.g. the discussion in [5, pg. 386-387].
Notation 2.6.
We denote by
Emb the class of embeddings between (abelian) groups.
Definition 2.7. (1) Let K be a class of groups and suppose that (K , Map ) and (K , Map ) are categories. Then we say that G ∈ K is (K , Map , Map ) -Hopfian(or (Map , Map ) -Hopfian) when f ∈ Map ( G, G ) implies f ∈ Map ( G, G ) .(2) We say that G ∈ K is co-Hopfian (resp. Hopfian) when G is (K , Map , Map ) -Hopfian, where K is the class of groups, Map is the class of embeddings (resp.onto homom.) and Map is the class of onto homom. (resp. embeddings).(3) More generally, when Map = Emb (cf. Not. 2.6), instead of (K , Map , Map ) -Hopfian we simply talk of Map -co-Hopfian groups (we do this in Theorem 1.9). GIANLUCA PAOLINI AND SAHARON SHELAH
Graphs and TreesDefinition 2.8.
By a directed graph we mean a structure in the language L = { R } ,where R is a binary predicate symbol. We say that the directed graph M is irreflexivewhen M | = ∀ x ( ¬ R ( x, x )) . We say that the directed graph M is asymmetric when M | = ∀ x ∀ y ( R ( x, y ) → ¬ R ( y, x )) . We say that the directed graph M has no cycles(or that it is acyclic) when there is no n < ω and x , ..., x n ∈ M such that: M | = x = x n , M | = R ( x n , x ) and, for every i < n , M | = R ( x i , x i +1 ) . Definition 2.9.
By a graph we mean a structure M in the language L = { R } ,where R is a binary predicate symbol, satisfying the following axioms:(i) ∀ x ( ¬ R ( x, x )) (irreflexivity of R );(ii) ∀ x ∀ y ( R ( x, y ) → R ( y, x )) (symmetry of R ).The graph M has no cycles when there is no n < ω and x , ..., x n ∈ M such that: M | = x = x n , M | = R ( x n , x ) and, for every i < n , M | = R ( x i , x i +1 ) . Definition 2.10.
Given an L -structure M by a partial automorphism of M wemean a partial function f : M → M such that f : h dom( f ) i M ∼ = h ran( f ) i M . Definition 2.11.
Let ( T, < T ) be a strict partial order.(1) ( T, < T ) is a tree when, for all t ∈ T , { s ∈ T : s < T t } is well-ordered by therelation < T . Notice that according to our definition a tree ( T, < T ) might havemore than one root, i.e. more than one < T -minimal element. We say that thetree ( T, < T ) is rooted when it has only one < T -minimal element (its root).(2) A branch of the tree ( T, < T ) is a maximal chain of the partial order ( T, < T ) .(3) A tree ( T, < T ) is said to be well-founded if it has only finite branches.(4) Given a tree ( T, < T ) and t ∈ T we let the level of t in ( T, < T ) , denoted as lev( t ) , to be the order type of { s ∈ T : s < T t } (recall item (1)). Concerning Def. 2.11(4), we will only consider trees (
T, < T ) such that, for every t ∈ T , { s ∈ T : s < T t } is finite, so for us lev( t ) will always be a natural number. Fact 2.12.
Let M be a graph, U 6 = V ⊆ M and assume that |U| = | M | = |V| = ℵ .Then the following are equivalent:(A) h is an isomorphism from M ↾ U onto M ↾ V ;(B) there is ¯ g = ( g k : k < ω ) such that:(a) for every k < ω , g k is a finite partial automorphism of M ;(b) for every k < ω , g k ( g k +1 ;(c) for every k < ω , g k = g − k ;(d) S k<ω g k = h . Borel Completeness of Torsion-Free Abelian Groups
The FrameHypothesis 3.1. (1) M is a graph with set of nodes ⊆ ω ;(2) G is the set of finite partial automorphisms g of M such that either dom( g ) = ∅ or g = g − . Notice that in particular G is closed under g g − ;(3) for m < ω , G m ∗ = { ( g , ..., g m − ) ∈ G m : g ( · · · ( g m − } . Notation 3.2. (1) We use s, t, ... to denote finite subsets of M and U , V , ... todenote arbitrary subsets of M . ORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE 7 (2) For ¯ g = ( g , ..., g lg(¯ g ) − ) ∈ G lg(¯ g ) ∗ and s, t ⊆ ω M , we let:(a) ¯ g [ s ] = t mean that S i< lg(¯ g ) g i [ s ] = t ;(b) dom(¯ g ) = dom( S i< lg(¯ g ) g i ) ;(c) ran(¯ g ) = ran( S i< lg(¯ g ) g i ) ;(d) ¯ g − = ( g − i : i < lg(¯ g )) Definition 3.3.
In the context of Hypothesis 3.1, let K bo1 ( M ) be the class of objects m ( M ) = m = ( X m , ¯ X m , I m , ¯ I m , ¯ f m , ¯ E m , ¯ p m , S m , < m ) = ( X, ¯ X, I, ¯ I, ¯ f, ¯ E, ¯ p, S, < ) such that the following conditions are satisfied:(1) X is a non-empty countable set and X ⊆ ω ;(2) (a) ( X ′ s : s ⊆ ω M ) is a partition of X into infinite sets;(b) for s ⊆ ω M , let X s = S t ⊆ s X ′ t ;(c) ¯ X = ( X s : s ⊆ ω M ) and so s ⊆ t ⊆ ω M implies X s ⊆ X t ;(3) for U ⊆ M let X U = S { X s : s ⊆ ω U} and so X = X M = S { X s : s ⊆ ω M } ;(4) (a) ¯ I = ( I n : n < ω ) = ( I m n : n < ω ) are pairwise disjoint;(b) ¯ g ∈ I n implies ¯ g ∈ G m ∗ for some m n ;(c) I n is finite;(5) if ¯ g ′ ⊳ ¯ g ∈ I n , then ¯ g ′ ∈ I
Let m ∈ K bo1 ( M ) . Notice that conditions (7a) and (17) of Defini-tion 3.3 imply that for every f ¯ g ∈ ¯ f m we have that dom( f ¯ g ) ∩ ran( f ¯ g ) = ∅ . Why?Suppose there is x ∈ dom( f ¯ g ) ∩ ran( f ¯ g ) , and let x ∈ X be such that f ¯ g ( x ) = x and x := f ¯ g ( x ) . Then ( x , x ) E ( x , x ) , contradicting Definition 3.3(17). Observation 3.5.
Let m ∈ K bo1 ( M ) . The set of conditions (1)-(23) from Defini-tion 3.3 is not minimal. In particular clauses (18)-(23) imply (16) and also: ( · ) if ¯ x , ..., ¯ x i ∗ − ∈ ¯ x/E n are pairwise distinct, then there exists j < i ∗ and ℓ < n such that x jℓ is < m -maximal in { x j , ..., x jn − } and the following holds: x jℓ / ∈ { x im : i < i ∗ , m < n, ( i, m ) = ( j, ℓ ) } . Proof.
For n = 1, both (16) and ( · ) are trivial. Let then n > j < i ∗ be such that ¯ x j is locally < m n -maximal (i.e., i < i ∗ implies ¯ x j < m n ¯ x i ).Let ℓ < n be such that x jℓ is < m -maximal in ¯ x j (recall that ¯ x j is with norepetitions). We claim that ( j, ℓ ) is as required in ( · ).We prove (+). Let ( i, m ) be a counterexample, i.e. i < i ∗ , m < n , ( i, m ) = ( j, ℓ ) and x im = x jℓ . By Definition 3.3(17), m = ℓ , so necessarily i = j and let ¯ y := ¯ x i ∧ m n ¯ x j .If ¯ y = ¯ x j , then, noticing firstly that ¯ y < m n ¯ x j we may apply Definition 3.3(23b) with¯ x i , ¯ x j , ¯ y here standing for ¯ x, ¯ y, ¯ z there, and so we have that x jℓ is not < m -maximal in ORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE 9 ¯ x j , a contradiction. If on the other hand ¯ y = ¯ x j , then ¯ x j = ¯ y < m n ¯ x i , as ¯ x j = ¯ y = ¯ x i cannot happen, but this contradicts the fact that i < i ∗ implies ¯ x j < m n ¯ x i .We are then left with proving (16). Now, if { ¯ x i : i < i ∗ } has two < m n -incomparableelements, then we are done by (+). Indeed, let i (1) , i (2) < i ( ∗ ) be such that¯ x i (1) , ¯ x i (2) are < m n -incomparable. Now, for ℓ ∈ { , } there is j ( ℓ ) < i ( ∗ ) suchthat ¯ x i ( ℓ ) m n ¯ x j ( ℓ ) and ¯ x j ( ℓ ) is locally < m n -maximal among { ¯ x i : i < i ( ∗ ) } . Wecan then choose m ( ℓ ) < n such that x j ( ℓ ) m ( ℓ ) is < m -maximal in ¯ x i ( ℓ ) . By (+) weknow that ( j ( ℓ ) , m ( ℓ )) are as required for ( · ). But then, by the choice of i (1) , i (2)and j (1) , j (2), also ¯ x j (1) , ¯ x j (2) are < m n -incomparable (by Definition 3.3(22)), hence j (1) = j (2). It follows that ( j (1) , m (1)), ( j (2) , m (2)) are as required for (16), andso in this case we are done. So we are left with the case in which { ¯ x i : i < i ∗ } is < m n -linearly ordered. W.l.o.g. we have the following situation:¯ x < m n ¯ x < m n · · · < m n ¯ x i ∗ − . By Def. 3.3(23c) the sets (ran(¯ x i ) : i < i ∗ ) are pairwise distinct, so we are done. Definition 3.6.
For m ∈ K bo1 ( M ) , we say that m ∈ K bo2 ( M ) when: ( ∗ ) X m = Y m ; ( ∗ ) if s ⊆ ω M , then for some x = y ∈ X ′ s we have ¬ (( x ) E ( y )) (this conditionactually follows by Definition 3.3(24) but we chose to include it for clarity); ( ∗ ) I = S n<ω I n = S m<ω G m ∗ (cf. Hypothesis 3.1(3)); ( ∗ ) if, for every n < ω , g n ∈ G and g n ( g n +1 , and U = S n<ω dom( g n ) ⊆ M ,then S n<ω dom( f ( g ℓ : ℓ For M as in Hypothesis 3.1, K bo2 ( M ) = ∅ . Proof. ( ∗ ) K bo0 ( M ) = ∅ .[Why? Let m be such that:(a) | X | = ℵ , and X ⊆ ω ;(b) ( X ′ s : s ⊆ ω M ) is a partition of X into infinite sets;(c) for s ⊆ ω M , X s = S t ⊆ s X ′ t ;(d) ¯ X = ( X s : s ⊆ ω M );(e) I m = { () } , f () is the empty function, ¯ f = ( f () ) and I n = ∅ , for every n < ω ;(f) S m = { () } .Notice that () denotes the empty sequence and under this choice of m , n ( m ) = 1,where we recall that the notation n ( m ) was introduced in Definition 3.7(1).]( ∗ ) If m ∈ K bo0 ( M ), n = n ( m ) > 0, ¯ g = ( g , ..., g m − ) ∈ I m 1, so ¯ g k +1 ↾ m ∈ I m k , and:( · . ) if m k is even, use ( ∗ ) with the pair n ( m k ), ¯ g k +1 here stand-ing for the pair n, ¯ g ⌢ ( g ) there;( · . ) if m k is odd, use ( ∗ ) with the pair n ( m k ), ¯ g k +1 here standingfor the pair n, ¯ g ⌢ ( g ) there.Clearly m = lim ℓ<ω ( m ℓ ) is as promised by the choice of (¯ g ℓ : ℓ < ω ), e.g. Def. 3.6( ∗ ) holds by Def. 3.7(2f), which in turn holds by ( ∗ . )( c ) of the present proof.Concerning < m , which is needed for Definition 3.3(18), let < m = S { < m ℓ : ℓ < ω } .As, by the first half of Definition 3.7(2h), ( Y m ℓ , < m ℓ ) is an increasing sequence oflinear orders, clearly ( Y m , < m ) is a linear order, and it is easy to see that it is oforder type ω , by the second half of Definition 3.7(2h). Finally, we show that Y m = X . It suffices to prove that, for any s ⊆ ω M , X ′ s ⊆ Y m . For this it suffices toprove that for any m < ω , { , ..., m − } ∩ X ′ s ⊆ Y m . But clearly for some ¯ g ∈ G ∗ m wehave that s ⊆ dom(¯ g ), and so for some ℓ > g ℓ = ¯ g , and also for some i > ℓ and g ′ , g ′′ ∈ G we have that ¯ g i = ¯ g ⌢ ( g ′ , g ′′ ). Thus by Definition 3.3(15) wehave that { , ..., m − } ∩ X ′ s ⊆ dom( f m ( i +1)¯ g ) ⊆ Y m . This concludes the proof. Definition 3.9. Let m ∈ K bo1 ( M ) .(1) Let G = G [ m ] be L { Q x : x ∈ X } .(2) Let G = G [ m ] be the subgroup of G generated by X , i.e. L { Z x : x ∈ X } .(3) Let G = G [ m ] be the subgroup of G generated by:(a) G ;(b) p − m ( P ℓ Let m ∈ K bo2 and ℓ ∈ { , , } .(1) G ℓ [ m ] ∈ TFAB and | G ℓ [ m ] | = ℵ .(2) Recalling ˆ f ¯ g = ˆ f g := ˆ f g ↾ G (1 , dom(¯ g )) (cf. Definition 3.9(5)(7)), we have thatthe map ˆ f ¯ g is a well-defined partial automorphism of G , and dom( ˆ f ¯ g ) is a puresubgroup of G [ m ] , in fact dom( ˆ f ¯ g ) is the pure closure in G of dom( ˆ f g ) .(3) If p = p ( e, ¯ q ) , e ∈ seq n ( X ) /E n , ¯ q = ( q ℓ : ℓ < n ) ∈ ( Z \ { } ) n and n > , then: G (1 ,p ) = h X { Z ( X ℓ Assume that m ∈ K bo2 ( M ) , U , V ⊆ M and |U| = |V| = ℵ .(1) If U 6 = V and M ↾ U ∼ = M ↾ V , then: G (1 , U ) [ m ] ∼ = G (1 , V ) [ m ] . (2) The following conditions are equivalent:(a) ( M ↾ U ∼ = M ↾ V and U 6 = V ) or ( U = V and there is an automorphism of M ↾ U which is not the identity on M ↾ U );(b) there is a sequence ( g k : k < ω ) such that, for every k < ω , g k ∈ G , g k ( g n +1 and S k<ω g k is an isomorphism from M ↾ U onto M ↾ V .Proof. We prove (1). Let h be an isomorphism from M ↾ U onto M ↾ V . Let( r ℓ : ℓ < ω ) list U with no repetitions (recall |U| = |V| = ℵ ) in a such a way that:(i) for k < ω , g k = h ↾ { r ℓ : ℓ k } ;(ii) ( g k : k < ω ) is as in Fact 2.12 with respect to h ;(iii) for k < ω , ¯ g k = ( g ℓ : ℓ k ), so ¯ g k ∈ G k +1 ∗ (cf. Hypothesis 3.1(3));(iv) s k = { r ℓ : ℓ k } = dom( g k ) and t k = { h ( r ℓ ) : ℓ k } = ran( g k );(v) by Definition 3.6( ∗ ) , for every k < ω we have that ¯ g k ∈ I m and so f ¯ g k ∈ ¯ f m .Notice now that for k < ω we have:( ⋆ ) (a) dom( f ¯ g k ) ⊆ X s k , ran( f ¯ g k ) ⊆ X t k ;(b) { , ..., k − } ∩ X s k ⊆ dom( f ¯ g k +1 );(c) { , ..., k − } ∩ X t k ⊆ dom( f − g k +1 ) = ran( f ¯ g k +1 ).[Why? (a) is by Def. 3.3(7b). (b) and (c) are by Definition 3.3(14)(15).]Notice also that:( ⋆ ) (d) S k<ω dom( f ¯ g k ) = S k<ω X s k = X U ;(e) S k<ω dom( f − g k ) = S k<ω X h [ s k ] = X V .[Why? (d) is by ( ⋆ )(a)-(b) and the fact that X ⊆ ω . (e) is by ( ⋆ )(a)-(b), the factthat X ⊆ ω and that h is from U onto V .]Hence, we have:( ⋆ ) S k<ω ˆ f ¯ g k is an isomorphism from G (1 , U ) onto G (1 , V ) (cf. Def. 3.9(7)).[Why? By Def. 3.9(5)(6)(7).] The proof of (2) is similar and anyhow not used.3.2. Analyzing IsomorphismHypothesis 3.12. Throughout this subsection the following hypothesis holds:(1) m ∈ K bo2 ( M ) ;(2) U 6 = V ⊆ M ;(3) |U| = ℵ = |V| ;(4) π is an isomorphism from G (1 , U ) [ m ] onto G (1 , V ) [ m ] . Lemma 3.13. Let a ∈ G (1 , U ) [ m ] \ { } and let b = π ( a ) .(1) For a prime p , a ∈ G (1 ,p ) ⇔ b ∈ G (1 ,p ) ;(2) if a = qx , for some q ∈ Q \ { } and x ∈ X U , then for some y ∈ X V :(a) ( x ) E ( y ) ;(b) b ∈ Q y , i.e. there exist m , m ∈ Z \ { } such that m b = m y .Proof. Item (1) is obvious by Hypothesis 3.12(4). Concerning item (2), let n < ω ,¯ y ∈ seq n ( X ) and ¯ q ∈ ( Q \ { } ) n be such that b = P { q ℓ y ℓ : ℓ < n } . It suffices toprove (2)(b), as if b = m m y let p ′ = p (( x ) /E , ¯ q ) , then x ∈ G (1 ,p ′ ) and so, by (1), y ∈ G (1 ,p ′ ) and thus by Lemma 3.10(3) we are done. Trivially, n > 0, we shall show ORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE 15 that n = 1, so toward contradiction we assume that n > 1. Let q ∗ ∈ ω \ { } besuch that b := q ∗ b ∈ G [ m ]. Let e = ¯ y/E n , q ′ ℓ = q ∗ q ℓ and ¯ q ′ = ( q ′ ℓ : ℓ < n ), so that q ∗ q ℓ y ℓ = q ′ ℓ y ℓ and q ′ ℓ ∈ Z \ { } . Let p = p ( e, ¯ q ′ ) . Then we have:( ∗ ) (i) b ∈ G (1 ,p ) ;(ii) a ∈ G (1 ,p ) .[Why (i)? By the choice of p we have that b ∈ G (1 ,p ) (cf. Def. 3.9(3)(4)) and so, as G (1 ,p ) is pure in G (cf. Observation 2.5), b = q b and q ∈ Z , we have b ∈ G (1 ,p ) (cf. Observation 2.4). Why (ii)? By (1) and ( ∗ )(i), recalling Hyp. 3.12(4).]By Lemma 3.10(3), there are k < ω , and, for i < k , ¯ y i ∈ ¯ y/E n and q i ∈ Q \ { } s.t.:( ∗ ) a = P i In the context of Conc. 3.14 and letting ( q x : x ∈ X U ) = ( q x : x ∈ X U ) .(1) For some q ∗ ∈ Q \ { } we have that, for every x ∈ X U , q x = q ∗ .(2) There is an isomorphism σ : M ↾ U ∼ = M ↾ V .Proof. Let x = y ∈ X U be such that ( x ) /E m = ( y ) /E m (cf. Definition 3.3(24)).Let then e = ( x, y ) /E , ¯ q = (1 , 1) and p = p ( e, ¯ q ) . Now, by the choice of p , we havethat x + y ∈ G (1 ,p ) and so q x π ( x ) + q y π ( y ) = π ( x ) + π ( y ) = π ( x + y ) ∈ G (1 ,p ) . So,by Lemma 3.10(3), there are ( x i , y i ) ∈ ( x, y ) /E and q i ∈ Q \ { } , for i < k , s.t.:( ⋆ ) q x π ( x ) + q y π ( y ) = P i In the context of Claim 3.15, q ∗ is an integer.Proof. If not, then q ∗ = mk , for m ∈ Z \ { } and k ∈ ω \ { , } . Let p be a primedividing k . Let x ∈ X U . If in G we have that x is not divisible by p , then weare done (since then π ( x ) cannot be q ∗ x ). Thus, by Lemma 3.10(3)(4), it mustbe the case that p = p ( x /E , ( q )) , for some q ∈ Q \ { } such that qx ∈ G , butby Definition 3.6( ∗ ) we can find x ∈ X U such that ( x ) / ∈ ( x ) /E , and so, byLemma 3.10(3), also in this case we reach a contradiction. Thus, q ∗ ∈ Z . Claim 3.17. In the context of Claim 3.15, q ∗ ∈ { , − } .Proof. If not, then we contradict Claim 3.16 when applied to π − .3.3. The Proof of the Main Theorem Notice that in this subsection Hypothesis 3.12 is no longer assumed. Conclusion 3.18. (1) Let m [ M ] ∈ K bo2 , U , V ⊆ M and |U| = |V| = ℵ . Then: ( ⋆ ) M ↾ U ∼ = M ↾ V ⇔ G (1 , U ) [ m ] ∼ = G (1 , V ) [ m ] . (2) Further, G (1 , U ) [ m ] has only trivial automorphisms (i.e. id G (1 , U ) [ m ] and g 7→ − g ).(3) In the context of Claim 3.11(2), clauses (a) and (b) are equivalent to:(c) there is an isomor. π from G (1 , U ) onto G (1 , V ) which = id G (1 , U ) , − id G (1 , U ) .Proof. We prove (1). First assume that U = V , then clearly both the left-hand-side(LHS) and the right-hand-side (RHS) of ( ⋆ ) holds. Assume then that U 6 = V . If theLHS of ( ⋆ ) holds, then by Claim 3.11(1) also the RHS of ( ⋆ ) holds. On the otherhand, if the RHS of ( ⋆ ) holds, then the assumptions in Hypothesis 3.12 are fulfilledand thus 3.13-3.17 holds, so in particular Claim 3.15(2) holds, and thus the LHS of( ⋆ ) holds. The proofs of items (2)-(3) and similar, and anyhow (3) is not used. Proof of Main Theorem. Let M be a saturated graph of cardinality ℵ , i.e. a copyof the universal homogeneous graph of size ℵ , and assume further that M hasdomain ω . Fix m ∈ K bo2 ( M ) (cf. Claim 3.8) and assume without loss of generalitythat G [ m ] has set of elements ω . For every graph H with domain ω we define F [ H ] : H → M by defining F [ H ]( n ) by induction on n < ω as the minimal k < ω ORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE 19 such that { ( ℓ, F [ H ]( ℓ )) : ℓ < n } ∪ { ( n, k ) } . Clearly, H is a graph isomorphismfrom H ↾ ( n + 1) onto M ↾ ( { F [ H ]( ℓ ) : ℓ < n } ∪ { k } ). Thus, the map: B : H G (1 , { F [ H ]( n ) : n<ω } ) [ m ]is a Borel map from Graph ω into TFAB ω and so by Conclusion 3.18 we are done.4. The Co-Hopfian Problem for Torsion-Free Abelian Groups Fact 4.1 ([4, Proposition 2.2, pg. 130]) . For G ∈ TFAB , G is co-Hopfian iff G isdivisible and of finite rank, i.e., G is a finitely dimensional vector space over Q . Conclusion 4.2. The co-Hopfian groups in TFAB ω form a Borel subset of TFAB ω . On the other hand, we will show below that there are variations on the notionof co-hopfianity (cf. Definition 2.7) which give a completely different answer. Hypothesis 4.3. Throughout this section the following hypothesis stands:(1) T = ( T, < T ) is a rooted tree with ω levels and we denote by lev( t ) the level of t ;(2) T = S n<ω T n , T n ⊆ T n +1 , and t ∈ T n implies that lev( t ) n ;(3) T = ∅ , T n is finite, and we let T Let K co1 ( T ) be the class of objects: m ( T ) = m = ( X Tn , ¯ f Tn : n < ω ) = ( X n , ¯ f n : n < ω ) satisfying the following requirements:(a) X = ∅ , X n is finite and strictly increasing with n , and X In the context of Definition 4.4, we have:(1) If m < n < ω , t ∈ T n \ T For T as in Hypothesis 4.3, K co1 ( T ) = ∅ (cf. Definition 4.4).Proof. Straightforward. Definition 4.8. On X (cf. Convention 4.5) we define:(1) for x ∈ X , suc( x ) = { f t ( x ) : t ∈ T, x ∈ dom( f t ) } ;(2) for x, y ∈ X , we let x < X y if and only if for some < n < ω and x , ..., x n ∈ X we have that V ℓ Let m ∈ K co1 ( T ) (i.e. as in Convention 4.5).(1) Let G = G [ m ] be L { Q x : x ∈ X } .(2) Let G = G [ m ] be the subgroup of G generated by X , i.e. L { Z x : x ∈ X } .(3) For t ∈ T , let:(a) H (2 ,t ) = L { Q x : x ∈ dom( f t ) } ; ORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE 21 (b) I (2 ,t ) = L { Q x : x ∈ ran( f t ) } ;(c) ˆ f t is the (unique) isomorphism from H (2 ,t ) onto I (2 ,t ) such that x ∈ dom( f t ) implies that ˆ f t ( x ) = f t ( x ) (cf. Definition 4.4(c)).(4) For t ∈ T , we define H (0 ,t ) := H (2 ,t ) ∩ G and I (0 ,t ) := I (2 ,t ) ∩ G ;(5) For ˆ f t as above, we have that ˆ f t [ H (0 ,t ) ] = I (0 ,t ) . We define ˆ f t as ˆ f t ↾ H (0 ,t ) .(6) We define the partial order < ∗ on G +0 := G \ { } by letting a < ∗ b if and onlyif a = b ∈ G +0 and, for some < n < ω , a , ..., a n ∈ G , a = a, a n = b and: ℓ < n ⇒ ∃ t ∈ T ( ˆ f t ( a ℓ ) = a ℓ +1 ) . (7) For a = P ℓ If (A), then (B), where:(A) (a) a, b ℓ ∈ G , for ℓ < ℓ ∗ ;(b) a ∗ b ℓ and the b ℓ ’s are with no repetitions;(c) a = P { q i x i : i < j } ;(d) ¯ x = ( x i : i < j ) ∈ X j is injective and reasonable;(e) q i ∈ Z \ { } ;(B) there are ℓ ∗ and for ℓ < ℓ ∗ , ¯ y ℓ = ( y ( ℓ,i ) : i < j ) such that:(a) y ( ℓ,i ) =: y ℓi ∈ X and ¯ x j ∗ ¯ y ℓ (cf. Definition 4.8(5));(b) b ℓ = P { q i y ( ℓ,i ) : i < j } , and so the ¯ y ℓ are pairwise distinct;(c) ( y ( ℓ,i ) : i < j ) is injective and reasonable;(d) if j > and ℓ ∗ > , then there are at least two y ∈ X such that: |{ ( ℓ, i ) : ℓ < ℓ ∗ , i < j and y ( ℓ,i ) = y }| = 1; (e) if j > and ℓ ∗ > , then there are ℓ = ℓ < ℓ ∗ and i , i < j such that:(i) if ℓ < ℓ ∗ , i < j and y ( ℓ,i ) = y ( ℓ ,i ) , then ( ℓ, i ) = ( ℓ , i ) ;(ii) if ℓ < ℓ ∗ , i < j and y ( ℓ,i ) = y ( ℓ ,i ) , then ( ℓ, i ) = ( ℓ , i ) .(f ) ( y ( ℓ,j − : ℓ < ℓ ∗ ) is without repetitions and none of { y ( ℓ,i ) : ℓ < ℓ ∗ , i By the definition of ∗ there are ( y ( ℓ,i ) : i < j, ℓ < ℓ ∗ ) satisfying clauses( a )-( c ) of ( B ) as in the proof of Lemma 4.11(7). Recall that ( { ¯ y : ¯ x jX ¯ y } , jX ) isa tree. We now imitate the proof of Observation 3.5.Case 1. { ¯ y ℓ : ℓ < ℓ ∗ } is not linearly ordered by jX .Then there are ℓ (1) = ℓ (2) < ℓ ∗ such that ¯ y ℓ (1) , ¯ y ℓ (2) are locally jX -maximal. Soas in the analogous case in the proof of Obs. 3.5 we can choose i , i < j s.t.: x ℓ i ∈ X n [ b ℓ ] \ X 1, ¯ y ℓ < jX ¯ y ℓ +1 . Now, for ℓ < ℓ ∗ and i < j , let n ( ℓ, i ) < ω be such that y ℓi ∈ X n ( ℓ,i ) \ X Let ( p a : a ∈ G +0 ) be a sequence of pairwise distinct primes.(1) For a ∈ G +0 , let: P ∗ a = { p b : b ∈ G +0 , b ∗ a } and P > ∗ a = { p b : b ∈ G +0 , a ∗ b } . (2) Let G = G [ m ] = G [ m ( T )] = G [ T ] be the subgroup of G generated by: { m − a : a ∈ G +0 , m ∈ ω \{ } a product of primes from P ∗ a , poss. with repetitions } . (3) For a prime p , let G (1 ,p ) = { a ∈ G : a is divisible by p m , for every < m < ω } (notice that, by Observation 2.5, G (1 ,p ) is always a pure subgroup of G ).(4) For b ∈ G +1 , let P b = { p a : a ∈ G +0 , G | = V m<ω p ma | b } . Remark 4.14. (1) If a, b ∈ G +1 and Q a = Q b ⊆ G , then P a = P b .(2) If b ∈ G +1 , then P b is infinite.Proof. Concerning (1), let q ∗ a = q ∗ b , where q ∗ , q ∗ ∈ Q \{ } . W.l.o.g. q ∗ , q ∗ ∈ Z \{ } and so q ∗ a = q ∗ b ∈ G . Let now p be an arbitrary prime, and, for ℓ ∈ { , } , let m ( ℓ ) < ω be such that that q ℓ = p m ( ℓ ) q ∗ ℓ , p q ∗ ℓ and ( q ∗ ℓ , p ) = 1. By transitivity ofequality, w.l.o.g. a ∈ G +0 . Now, let m ∈ Z with m > 0. Then we have:(a) In G , p m | a iff p m | q ∗ a .[Why (a)? First assume that G | = p m | a , then there is a ∈ G such that G | = p m a = a . Let a = q ∗ a , then G | = q ∗ a = q ∗ ( p m a ) = p m ( q ∗ a ) = p m a , so G | = p m | q ∗ a . Assume now that G | = p m | q ∗ a , and let q ∗ a = p m a , with a ∈ G .By the choice of p we know that ( p, q ∗ ) = 1 and so also ( p m , q ∗ ) = 1. It followsthat 1 belongs to the ideal of Z that p m and q ∗ generates, hence:for some m , m ∈ Z , we have m p m + m q ∗ = 1 . ORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE 23 But then: a = 1 · a = ( m p m + m q ∗ ) a = m p m a + m q ∗ a = p m ( m a ) + ( m p m a )= p m ( m a ) + p m ( m a )= p m ( m a + m a ) , and so p m | a , and thus we are done proving item (a).](b) In G , p m | q ∗ a iff p m + m (1) | p m (1) q ∗ a .[Why (b)? Similar to (a).](c) In G , p m + m (1) | p m (1) q ∗ a iff p m + m (1) | p m (2) q ∗ b .[Why? Since by our assumptions, p m (1) q ∗ a = p m (2) q ∗ b .](d) In G , p m + m (1) | p m (2) q ∗ b iff p m + m (1) − m (2) | q ∗ b .[Why? Like (b).](e) In G , p m + m (1) − m (2) | q ∗ b iff p m + m (1) − m (2) | b .[Why? Like (a).]Thus, putting everything together we have:(f) In G , p m | a iff p m + m (1) − m (2) | b .As for n < ω we have p n +1 | c implies p n | c , clearly: ^ n<ω p m | a ⇔ ^ n<ω p m | b. As p was an arbitrary prime, this concludes the proof of (1). Also, item (2) followsfrom (1) considering the distinct primes p b , p b , p b , ... . Lemma 4.15. (1) If p = p a , a ∈ G +0 , then: G (1 ,p ) = h b ∈ G +0 : a ∗ b i ∗ G . (2) For t ∈ T , H (1 ,t ) := H (2 ,t ) ∩ G and I (1 ,t ) := I (2 ,t ) ∩ G are pure in G .(3) For ˆ f ( i, t as in Definition 4.10(3c), ˆ f ( i, t [ H (1 ,t ) ] ⊆ I (1 ,t ) . We define ˆ f ( i, t as ˆ f ( i, t ↾ H (1 ,t ) .(4) ˆ f t ↾ H (1 ,t ) = I (1 ,t ) .Proof. Item (1) is clear by Claim 4.13(1)(2). Concerning item (2), simply notice: H (1 ,t ) = h Z x : x ∈ dom( f t ) i ∗ G ,I (2 ,t ) = h Z x : x ∈ ran( f t ) i ∗ G . Item (3) is by item (2) and the following observation, if f t ( x ) = y , then we have x ∗ y (recall Lemma 4.11(2)), and so P x ⊆ P y (cf. Definition 4.13(1)). Finally,concerning item (4), assume that 0 < n < ω and t ∈ T n \ T Let m ( T ) ∈ K co1 ( T ) .(1) We can modify the construction so that G [ m ( T )] = G [ T ] has domain ω andthe function T G [ T ] is Borel (for T a tree with domain ω ).(2) T has an infinite branch iff G [ T ] is not Emb -co-Hopfian.(3) T has an infinite branch iff G [ T ] is not Emb -co-Hopfian.(4) T has an infinite branch iff G [ T ] is not Emb -co-Hopfian.Proof. Item (1) is easy. We prove items (2)-(4) with a single proof. Concerningthe “left-to-right” direction of items (2)-(4), let ( t n : n < ω ) be an infinite branchof T . By Lemma 4.11(4), ( ˆ f t n : n < ω ) is increasing, by Definition 4.10(3c), ˆ f t n embeds H (2 ,t n ) into I (2 ,t n ) , thus ˆ f = S n<ω ˆ f t n is an embedding of G into G ,since G = S n<ω H (2 ,t n ) , where ( H (2 ,t n ) : n < ω ) is a chain of pure subgroups of G with limit G , because, recalling 4.4(e), we have that: H (2 ,t n ) ⊇ dom( f t n ) ⊆ dom( f t n +1 ) ⊆ H (2 ,t n +1 ) and by 4.4(c) we have that S n<ω H (2 ,t n ) = G . Thus ˆ f := ˆ f ↾ G = S n<ω ˆ f t n = S n<ω ˆ f t n ↾ H (1 ,t n ) is an embedding of G into G (cf. Lemma 4.15(3)), in fact wehave that dom( ˆ f t n ) = H (1 ,t n ) (cf. Lemma 4.15(3)) and G = S n<ω H (1 ,t n ) , where( H (1 ,t n ) : n < ω ) is chain of pure subgroups of G with limit G . Clearly ˆ f is notof the form g mg for some m ∈ Z \ { } , since for every x ∈ dom( f t ) we have x = f t ( x ) (cf. Obs. 4.6), this is enough for the “left-to-right” direction of item (4).We claim that G / ˆ f [ G ] is not torsion. To this extent, first of all notice that X = ∅ (by Definition 4.4(a)) and X ∩ ran( f t n ) = ∅ (by Definition 4.4(d)). Thus:ran( ˆ f ) ⊆ G X \ X := X { Q x : x ∈ X \ X } = h X \ X i ∗ G . Now, let x ∈ X , then x ∈ G \ ran( ˆ f ), moreover, for q ∈ Q \ { } : qx / ∈ G X \ X and so qx / ∈ ran( ˆ f ) , and so in particular, for every 0 < n < ω we have that nx / ∈ ran( ˆ f ), hence n ( x/ (ran( ˆ f )) = 0. This is enough for the “left-to-right” of items (2) and (3).We now prove the “right-to-left” direction of item (2). To this extent, supposethat ( T, < T ) is well-founded and, for the sake of contradiction, suppose that thereexists f ∈ End( G ) one-to-one such that G /f [ G ] is not torsion. Let G ∗ = G and G ∗ n +1 = f ( G ∗ n ) and notice that the sequence ( G ∗ n : n < ω ) is strictly ⊆ -decreasing.Let now c ∗ ∈ G ∗ be such that c ∗ /f [ G ∗ ] is not torsion in G ∗ /f [ G ∗ ], and let then,for 0 < n < ω , c ∗ n = f n ( c ∗ ), where f = f and f n +1 = f n ◦ f . Notice that then ORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE 25 for every n < ω , c ∗ n ∈ G ∗ n and c ∗ n /G ∗ n +1 is not torsion in G ∗ n /G ∗ n +1 . Thus, for every n < ω , c ∗ n ∈ G ∗ n ⊆ G ∗ = G ⊆ G , and so we have that:( ∗ ) c ∗ n = X { q ( n,x ) x : x ∈ w n } , where w n ⊆ X is finite and non-empty, and q ( n,x ) ∈ Q \ { } . Notice that:( ∗ ) (i) for n = m < ω we have that Q c ∗ n = Q c ∗ m (in G );(ii) S n<ω w n is infinite.[Why? Because ( c ∗ n : n < ω ) is linearly independent in the Q -vector space G . Tosee this, toward contradiction, suppose there is ℓ ∗ < ω and n (0) < · · · < n ( ℓ ∗ ) and q , ..., q ℓ ∗ ∈ Q \ { } such that G | = P ℓ ℓ ∗ q ℓ c ∗ n ( ℓ ) = 0. Then: X <ℓ ℓ ∗ q ℓ c ∗ n ( ℓ ) ∈ G ∗ n (0)+1 , and so q c ∗ n (0) ∈ G ∗ n (0)+1 , contradicting that c ∗ n (0) /G ∗ n (0)+1 is not torsion, as letting q = n /m , with n , m ∈ ω \ { } , m q ∈ Z \ { } and ( m q ) c ∗ n (0) ∈ G ∗ n (0)+1 .]Notice now that:( ∗ ) for every n < ω , there is h n : w n +1 → w n such that y ∈ w n +1 ⇒ h n ( y ) X y .[Why? Fix n < ω , then, by the definition of G (1 ,p ) (Definition 4.13(3)) and thechoice of ( f m : m < ω ) and ( c ∗ m : m < ω ), for every prime p , we have:( ∗ . ) c ∗ n ∈ G (1 ,p ) ⇒ c ∗ n +1 ∈ G (1 ,p ) . Let m n ∈ ω \ { } be such that m n c ∗ n := c + n ∈ G and let p ′ = p c + n , then c + n ∈ G (1 ,p ′ ) , and so, since G (1 ,p ′ ) is pure in G (cf. Observation 2.5), m n c ∗ n = c + n and m n ∈ Z \ { } , we have c ∗ n ∈ G (1 ,p ′ ) (cf. Observation 2.4). Thus, by ( ∗ ) andLemma 4.15(1), there is k ∈ Z \ { } such that the following holds:( ∗ . ) kc ∗ n +1 ∈ X { Z b : c ∗ n ∗ b } . Hence, there are j < ω , c ∗ n ∗ b , ..., b j − ∈ G +0 , and k, k , ..., k j − ∈ Z \ { } s.t.:( ∗ . ) G | = kc ∗ n +1 = X i X, < X ), contradicting the fact that ( X, < X ) is awell-founded tree (cf. Observation 4.9(2) recalling that T is well-founded). Thus,we have finished proving item (2) of the present theorem.We now prove the “right-to-left” direction of items (3)-(4). To this extent, relyingon the “right-to-left” direction of item (2), it suffices to show that if f ∈ End( G )is one-to-one and G /f [ G ] is torsion, then:(a) G /f [ G ] is bounded;(b) for some m ∈ Z \ { } we have that f ( a ) = ma , for all a ∈ G .Since G /f [ G ] is torsion, for each x ∈ X , there is m x ∈ Z \ { } such that m x x ∈ ran( f ). Fix now x ∈ X . Then we can find a ∈ G +1 such that f ( a ) = m x x , further,as we can replace the pair ( a, m ) by the pair ( ma, mm x ) for any m ∈ Z \ { } , wecan assume w.l.o.g. that a ∈ G +0 . We claim that:( ⋆ ) a ∈ Q x .To this extent, let p = p a . Then a ∈ G (1 ,p ) , and so f ( a ) = m x x ∈ G (1 ,p ) . Thus,since G (1 ,p ) is pure in G (cf. Lemma 4.15(1)), we have that x ∈ G (1 ,p ) (cf.Observation 2.4). But then, again by Lemma 4.15(1), we can find n < ω and m , m (2 , , ..., m (2 ,n − ∈ Z \ { } , and b , ..., b n − ∈ G +0 such that:( ⋆ . ) (i) m x = P { m (2 ,ℓ ) b ℓ : ℓ < n } ∈ G ;(ii) a ∗ b ℓ , for every ℓ < n ;(iii) the b ℓ ’s are pairwise distinct (w.l.o.g.).Suppose now (as a ∈ G +0 ) that a = P { q j y j : j < j ∗ } for some j ∗ < ω , q , ..., q j ∗ − ∈ Z \ { } , and y j ∈ X , with ( y , ..., y j − ) without repetitions. Clearly j ∗ > ⋆ . ) We claim that j ∗ = 1.For the sake of contradiction suppose that j ∗ > 1. Now, as for every ℓ < n , a ∗ b ℓ ,there are ( ˆ f t ( ℓ,i ) : i < i ( ℓ )) such that f ℓ = ˆ f t ( ℓ,i ( ℓ ) − ◦ · · · ◦ ˆ f t ( ℓ, and f ℓ ( a ) = b ℓ .Thus, by ( ⋆ . )(i), we have:( ⋆ . ) m x = X ℓ 2. Then, w.l.o.g. ( y j : j < j ∗ ) is reasonable (cf.Lemma 4.9(6)), and so using Claim 4.12 (with ( y j : j < j ∗ ) here as ( x i : i < j )there) we immediately get a contradiction, since the support of the right hand sideof ( ⋆ . ) has at least 2 members by Claim 4.12(B)(d)-(e), while the support of theleft hand side of ( ⋆ . ) has exactly one member. Thus, ( ⋆ . ) holds, as wanted. ORSION-FREE ABELIAN GROUPS ARE BOREL COMPLETE 27 Let ˆ f be the extension of f to an embedding of G into G (which exists as f embeds G into G and G ⊆ G = h G i ∗ G ). Hence, by ( ⋆ ) we have:( ⋆ ) for every x ∈ X there is q x ∈ Q such that G | = ˆ f ( q x x ) = x .Furthermore, we have:( ⋆ ) the sequence ( q x : x ∈ X ) is constant, call it q ∗ .[Why? Toward contradiction, suppose that x = x ∈ X and q x = q x . Then wehave x + x ∈ G +0 and f ( x + x ) = q x x + q x x . 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