Orthogonal decomposition of definable groups
Alessandro Berarducci, Pantelis E. Eleftheriou, Marcello Mamino
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ALESSANDRO BERARDUCCI, PANTELIS E. ELEFTHERIOU, AND MARCELLO MAMINOAbstract. Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat oppositenotion of cohesiveness, capturing the idea of interaction among all parts of agiven definable set. A cohesive set is indecomposable, in the sense that if it isinternal to the product of two orthogonal sets, then it is internal to one of thetwo. We prove that a definable group in an o-minimal structure is a product ofcohesive orthogonal subsets. If the group has dimension one, or it is definablysimple, then it is itself cohesive. Finally, we show that an abelian group definablein the disjoint union of finitely many o-minimal structures is a quotient, by adiscrete normal subgroup, of a direct product of locally definable groups in thesingle structures.
Contents1. Introduction 12. Orthogonality 43. Cohesive sets 74. Splitting and decomposition 95. Compact domination 116. Decomposition of compactly dominated abelian groups (NIP) 127. Decomposition of definably compact abelian groups (o-minimal) 138. Decomposition: general case (o-minimal) 149. Locally definable groups 1710. Questions 1811. Appendix 18References 201. IntroductionConsidering a group G interpretable in the disjoint union of finitely many structures X , . . . , X n (seen as a multi-sorted structure as in Definitin 9.2), one may ask whether G can be understood in terms of groups definable in the individual structures. Onemay, for instance, ask whether G is definably isomorphic to a quotient, modulo a Date : January 3, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Model theory, definable groups, o-minimality, NIP theories.Berarducci and Mamino have been partially supported by the Italian research project PRIN 2017,“Mathematical logic: models, sets, computability”, Prot. 2017NWTM8RPRIN. Eleftheriou has beensupported by 2017NWTM8RPRIN and a Zukunftskolleg Research Fellowship (Konstanz). finite normal subgroup Γ, of a direct product G × . . . × G n , where G i is a groupdefinable in X i . This however is not true in general: a counterexample is providedby [2, Example 1.2] (a torus obtained from two orthogonal copies of R and a latticegenerated by two vectors in generic position). After a talk by the first author atthe Oberwolfach workshop “Model Theory: Groups, Geometry, and Combinatorics”(2013), Hrushovski suggested that a result of the above kind would require to passto the locally definable category. So the natural conjecture would be that, if G is asabove, there is a locally definable isomorphism G ∼ = G × . . . × G n / Γ, where G i is alocally definable group in X i and Γ is a compatible discrete subgroup (i.e. a subgroupwhich intersects every definable set at a finite set). Here we extablish the conjectureunder the additional assumption that the structures X i are o-minimal and G is abelian.More precisely, we prove the following result. Theorem (9.3) . Let G be a definable abelian group in the disjoint union of finitelymany o-minimal structures X , . . . , X n . Then there is a locally definable homomor-phism G ∼ = G × . . . × G n / Γ , where G i is a locally definable group in X i and Γ is a compatible locally definablediscrete subgroup of G × . . . × G n . We will deduce Theorem 9.3 from Theorem 8.4 below, which is interesting in itselfand holds for non-abelian groups as well. To state the theorem we need to recall themodel theoretic notion of orthogonality. Given definable sets X , . . . , X n in a structure M , we say that X , . . . , X n are orthogonal if, for all k , . . . , k n ∈ N , any definablesubset of X k × . . . × X k n is a finite union of sets of the form A × . . . × A n , where A i is a definable subset of X k i i . Let us also recall that a definable set X is internal toa definable set Y if there is m ∈ N and a definable surjective map from Y m to X . Theorem (8.4) . Assume M is an o-minimal structure. Let X , . . . , X n be orthogonaldefinable sets in M and G a group definable in M . If G is internal to the product X × · · · × X n , then G is a product G = A . . . A n of definable subsets A , . . . , A n , where A i is internal to X i , for i = 1 , . . . , n . To deduce Theorem 9.3 from Theorem 8.4, we take for M the o-minimal structureconsisting of the concatenation of the structures X i separated by single points. Thestructures X i would then be orthogonal within M , so we can apply Theorem 8.4 andtake as G i an isomorphic copy of the subgroup of G generated by A i . The only delicatepoint is to show that G i is locally definable in X i , but this is not difficult.It is worth stressing that Theorem 8.4 holds for an arbitrary o-minimal structure M and in particular we do not assume that M has a group structure. This is importantfor the application to Theorem 8.4 because in general the concatenation of o-minimalstructures does not have a group structure even if the single structures do.The proof of Theorem 8.4 uses a number of deep results about groups definable ino-minimal structures, such as the solution of Pillay’s Conjecture and Compact Dom-ination Conjecture (see [15] and [9] for the definitions). It is, however, conceivablethat the theorem could be extended far beyond the o-minimal context: indeed we have RTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS 3 no counterexample even if M is allowed to be a completely arbitrary structure. The-orem 6.1 provides a partial result in this direction, when M is NIP and G is compactlydominated and abelian.Before stating our final result, we notice that the sets A , . . . , A n in Theorem8.4 are orthogonal, so we may ask whether, for a group G definable in an arbitraryo-minimal structure M , there is always a natural way to generate it as a product G = A . . . A n of orthogonal definable subsets A i . Of course there is always the trivialsolution with n = 1 and A = G , but we would like each A i to be in some sense“minimal”. To this aim we introduce the following model theoretic notions. Let Z bea set definable in an arbitrary structure M . We say that: • Z is indecomposable if whenever Z is internal to the cartesian product oftwo orthogonal sets, then Z is internal to one of the two. • Z is cohesive if whenever two definable sets are non-orthogonal to Z , theyare non-orthogonal to each other.A cohesive set Z is indecomposable (Proposition 3.2(2)), but we do not know whetherthe converse holds. We show that cohesive sets have nice model-theoretic propertieswhich we are not able to prove for indecomposable sets. In particular, a set internal toa cohesive set is cohesive and the cartesian product of two cohesive sets is cohesive.The intuition is that the various parts of a cohesive set interact in a non-trivial way, soin particular a cohesive set cannot contain two orthogonal infinite definable subsets.Our final result is the following. Theorem (8.7) . Let G be a group interpretable in an o-minimal structure M . Thenthere are cohesive orthogonal definable sets A , . . . , A n , such that G = A . . . A n . In the setting of Theorem 8.7 we call the tuple A , . . . , A n an orthogonal decom-position of G , while in Theorem 8.4 the tuple A , . . . , A n is called a decomposition of G with respect to X , . . . , X n . One can loosely make the following analogy: a decom-position is like a factorization, while an orthogonal decomposition is like a factorizationinto primes.If G is infinite, we can choose each A i in Theorem 9.3 to be infinite, and in thiscase the number n is an invariant of G up to definable isomorphism. Indeed, if G = B . . . B m is another decomposition of G as a product of orthogonal cohesiveinfinite definable subsets, then m = n and each B i is bi-internal to a single A j . We maycall the invariant n the dimensionality of G (not to be confused with the dimensionof G ). In this terminology, the unidimensional groups in the sense of [13, Claim 1.26]have dimensionality 1.We can show that if G has dimension one, or it is definably simple, then G is itselfcohesive, so these groups have dimensionality one.For the proof of Theorem 8.7, we need both Theorem 8.4 and the results from [7].In particular, we need that for every group G interpretable in an o-minimal structure,there is an injective definable map f (not necessarily a morphism) from G to thecartesian product of finitely many 1-dimensional definable groups ([7, Theorem 3]). Related work.
Groups definable in the disjoint union of orthogonal structures havealready been considered in [2, 19]. The original motivation comes from the modeltheory of group extensions [3, 20]. For instance, the universal cover of a groupdefinable in an o-minimal expansion of the field R is definable in the disjoint union ALESSANDRO BERARDUCCI, PANTELIS E. ELEFTHERIOU, AND MARCELLO MAMINO R ⊔ Z , where Z has only the additive structure [10]. From a model-theoretic pointof view, ( Z , +) is an example of a superstable structure of finite and definable Lascarrank. In [2], it is shown that if a group G is definable in the disjoint union of anarbitrary structure R and a superstable structure Z of finite and definable Lascarrank, then G is an extension of a group internal to R by a group internal to Z .In [19], Wagner weakened the superstability assumption to assuming only that Z issimple. The simplicity assumption cannot be entirely removed, or replaced by an o-minimality one: quotients by a lattice of a product of copies of R provide examplesof definable groups which may not have infinite definable subgroups internal to anyof the copies ([2, Example 1.2]). However, as Theorem 8.4 shows, the o-minimalityassumption allows for another, in fact more symmetric analysis in terms of generatingsubsets instead of subgroups. Theorem 8.7 can also be seen as a continuation of thework in [7].The possibility of extending the results beyond the o-minimal context raises anumber of questions, which are included in Section 10. Structure of the paper.
In Sections 2 – 4, we introduce and study the key notionsof this paper. In particular we prove, for definable sets X , . . . , X n in an arbitrarystructure, that X , . . . , X n are orthogonal if and only if they are pairwise orthogonal.We then study indecomposable and cohesive sets and establish that 1-dimensionalgroups definable in o-minimal structures are cohesive (Theorem 3.4). The proofsof Theorems 8.4 and 8.7 then proceed in several steps. In Section 5, we recall thebasics of compact domination for NIP structures, which we employ in Section 6 toprove Theorem 8.4 for compactly dominated abelian NIP groups that are contained in X × · · · × X n (Theorem 6.1). In Section 7, we specialize in o-minimal structures andprove Theorem 8.4 for definably compact abelian groups internal to X × · · · × X n . InSection 8, we employ the rich machinery available for groups definable in o-minimalstructures, and conclude the full Theorem 8.4. Together with the results from [7],we then establish Theorem 8.7. In the final part of the paper we prove Theorem 9.3. Notation.
Throughout this paper, we work in a first-order structure M . By “de-finable” we mean definable in M , with parameters. Unless stated otherwise, X, Y, Z denote definable sets. By convention, X × Y = X .We assume familiarity with the basics of o-minimality, as in [5]. We also assumefamiliarity with the definable manifold topology of definable groups [14, Proposition2.5]. All topological notions for definable groups, such as connectedness and definablecompactness, are taken with respect to this group topology.2. OrthogonalityIn this section we work in an arbitrary structure M . We recall the notions oforthogonality and internality, and prove some basic facts. Definition 2.1.
Given definable sets X , . . . , X n , a ( X , . . . , X n ) -box is a definableset of the form U × · · · × U n where U i ⊆ X i for every i = 1 , . . . , n . When clear fromthe context, we omit the prefix ( X , . . . , X n )- in front of “box”. Definition 2.2.
Let X , . . . , X n be definable sets. We say that X , . . . , X n are or-thogonal if, for every k , . . . , k n ∈ N , every definable subset S of X k × . . . × X k n n isthe union of finitely many ( X k , . . . , X k n n )-boxes. RTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS 5
The definition of orthogonality can be rephrased in terms of types using the fol-lowing remark of Wagner. We include a proof to facilitate comparison with similarnotions of orthogonality in the model-theoretic literature (see [18]), but we will notneed this fact.
Remark 2.3 ([19, Remark 3.4]) . Let X , . . . , X n be definable sets. The followingconditions are equivalent:(1) Every definable subset of X ×· · ·× X n is a finite union of ( X , . . . , X n )-boxes.(2) For every type p ( x , . . . , x n ) over M with p ( x , . . . , x n ) ⊢ X × · · · × X n , wehave that p ( x ) ∪ · · · ∪ p n ( x n ) ⊢ p ( x , . . . , x n ) where p i is the i -th projectionof p . Proof.
To simplity notation we assume n = 2 and write X, Y for X , X .Assume (1). Let p ( x, y ) be a type over M concentrated on X × Y . Let ϕ ( x, y )be a formula over M defining a subset of X × Y . The set Z defined by ϕ is a finiteunion of boxes U i × V i . It follows that p ( x ) ∪ p ( y ) ⊢ p ( x, y ).Assume (2). Let Z ⊆ X × Y be definable. We must show that Z is a finiteunion of boxes U i × V i . For any type p ( x, y ) containing the defining formula of Z ,we have that p ( x ) ∪ p ( y ) ⊢ ( x, y ) ∈ Z . By compactness, there are ϕ p ( x ) ∈ p ( x )and ϕ p ( y ) ∈ p ( y ), such that ϕ p ( x ) ∧ ϕ p ( y ) ⊢ ( x, y ) ∈ Z . Again by compactness,( x, y ) ∈ Z is equivalent to a finite disjunction of formulas of the form ϕ p ( x ) ∧ ϕ p ( y )(if not we reach a contradiction considering a type containing the defining formula of Z and the negation of all the formulas ϕ p ( x ) ∧ ϕ p ( y )). It follows that Z is a finiteunion of boxes U i × V i as desired. (cid:3) The following fact follows at once from the definition.
Fact 2.4 ([2, Lemma 2.2]) . Let X and Y be orthogonal definable sets and let ( T s | s ∈ S ) be a definable family of subsets of a Y -internal set T indexed by an X -internalset S . Then the family contains only finitely many distinct sets T s ⊆ T . Proposition 2.5.
Let X , Y , Z be definable sets. Suppose that, for every positiveinteger n , all definable subsets of X n × Y are a finite union of ( X n , Y ) -boxes, and alldefinable subsets of X n × Z are finite unions of ( X n , Z ) -boxes. Then, for all n , everydefinable subset of X n × Y × Z is a finite union of ( X n , Y × Z ) -boxes.Proof. Let S be a definable subset of X n × Y × Z . Given z ∈ Z , consider the fiber S z ⊆ X n × Y consisting of the pairs ( x, y ) such that ( x, y , z ) ∈ S . Let E z ⊆ X n × X n be the following equivalence relation: a E z b ⇐⇒ ∀ y ∈ Y ( a, y ) ∈ S z ←→ ( b, y ) ∈ S z .Then { E z | z ∈ Z } is a family of subsets of X n × X n indexed by Z , so by thehypothesis applied to X n × Z , it is finite. By the hypothesis on X n × Y , eachequivalence relation E z has finitely many equivalence classes; in fact, S z is a finiteunion of ( X n , Y )-boxes and the equivalence classes of E z are the atoms of the booleanalgebra generated by the projections of these boxes on the component X n . It followsthat E = T z E z ⊆ X n × X n is again an equivalence relation with finitely many classes.On the other hand, for a, b ∈ X n , a E b ⇐⇒ ∀ y ∈ Y, z ∈ Z ( a, y , z ) ∈ S ←→ ( b, y , z ) ∈ S . ALESSANDRO BERARDUCCI, PANTELIS E. ELEFTHERIOU, AND MARCELLO MAMINO
We have thus proved that there are finitely many subsets of the form π Y × Z ( π − X n ( x ) ∩ S )with x ∈ X n , which is desired result. (cid:3) Corollary 2.6.
Let X , Y , Z be definable sets. If X is orthogonal to both Y and Z ,then X is orthogonal to Y × Z . Corollary 2.7.
Suppose X , . . . , X n are pairwise orthogonal definable sets. Then X , . . . , X n are orthogonal.Proof. If suffices to show that each X i is orthogonal to the product of the other sets X j . This follows by 2.6 and induction on n . (cid:3) We shall need the following result.
Corollary 2.8.
Let X and Y be definable sets. Then X and Y are orthogonal if andonly if for every positive integer n , all definable subsets of X n × Y are finite unions of ( X n , Y ) -boxes. For the main results of this paper we make no saturation assumptions on the ambi-ent structure M . It is however worth mentioning that under a saturation assumptionwe can strengthen Corollary 2.8 as follows. Proposition 2.9. If M is ℵ -saturated, then X and Y are orthogonal if and only ifall definable subsets of X × Y are finite unions of ( X, Y ) -boxes.Proof. Suppose that all definable subsets of X × Y are finite unions of ( X, Y )-boxes.Let S be a definable subset of X n × Y . It suffices to show that S is a finite unionof ( X n , Y )-boxes (by Corollary 2.8). We reason by induction on n . The case n = 1holds by the assumptions. Assume n >
1. For t ∈ X , consider the set S t = { ( x, y ) | ( t, x, y ) ∈ S } ⊆ X n − × Y. By induction, S t is a finite union of ( X n − , Y )-boxes. Let R t ⊆ Y × Y be theequivalence relation defined by uR t v ⇐⇒ ( ∀ x ∈ X ) ( x, u ) ∈ S t ↔ ( x, v ) ∈ S t .Note that R t has finitely many equivalence classes. By saturation, there is a uniformbound k ∈ N , such that for all t ∈ X , there are at most k equivalence classes modulo R t .We claim that there is a finite subset A of Y such that for all t ∈ X each equivalenceclass of R t intersects A . To this aim we prove, by induction on j ≤ k , that there is afinite subset A j of Y such that for all t ∈ Y there are at most k − j equivalence classesof R t which do not intersect A j . Clearly we can take A to be the empty set. Let usconstruct A j +1 . Consider the definable family ( S t ) t ∈ X where S t ⊆ Y is the union ofthe R t -equivalence class which intersect A j . By the assumptions the family ( S t ) t ∈ X contains finitely many distinct subsets S , . . . , S l of Y . We obtain A j +1 adding to A j a point in Y \ S i for each i = 1 , . . . , l such that S i = Y . The claim is thus provedtaking A = A k .Now we claim that the equivalence relation R = T t ∈ X R t has finitely many equiva-lence classes. Indeed, given a ∈ A consider the family ( P t,a ) t ∈ Y where P t,a ⊆ Y is the R t -equivalence class of a . For each a ∈ A , by the assumption there are finitely manysets of the form P t,a for t ∈ X . Each R t -equivalence class is of the form P t,a . Henceeach R equivalence class belongs to the finite boolean algebra generated by the sets P t,a . The claim is thus proved. RTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS 7
Now for each R -equivalence class B ⊆ Y and for each y , y ∈ B , we have ∀ t ∈ X, ∀ x ∈ X n − ( t, x, y ) ∈ S ⇐⇒ ( t, x, y ) ∈ S . Thus S ∩ π − Y B is a box where π Y : X n × Y → Y is the projection. Therefore S is a finite union of ( X n , Y )-boxes. (cid:3) We now turn to the notion of internality.
Definition 2.10 ([18, Lemma 10.1.4]) . Let M be a structure. Given two definablesets X and V , we say that X is internal to V , or V -internal , if there is a definablesurjection from V n to X , for some n . Proposition 2.11 ([2, Proposition 2.3]) . Let X and Y be orthogonal definable setsand S a definable subset of X × Y . Then S is X -internal if and only if its projectiononto Y is finite. It is clear that if one of X and Y is internal to the other, then the two sets arenon-orthogonal. Also, if one of X and Y is finite, then the two sets are orthogonal. Proposition 2.12.
Let X and Y be orthogonal definable sets. If U is internal to X and V is internal to Y , then U and V are orthogonal.Proof. Let S ⊆ U k × V l be definable. By the assumption there are definable surjectivemaps f : X m → U k and g : Y n → V l .Then, by orthogonality, ( f × g ) − ( S ) is a finiteunion S i A i × B i of ( X m , Y n )-boxes. So S = S i f ( A i ) × g ( B i ) is a finite union of( U k , V l )-boxes. (cid:3)
3. Cohesive setsIn this section we work in an arbitrary structure M , except for Theorem 3.4 wherewe assume o-minimality. We introduce and study the following two key notions, alsomentioned in the introduction. Definition 3.1.
Let Z be a definable set. • Z is indecomposable if for every orthogonal definable sets X, Y , if Z isinternal to X × Y , then Z is either internal to X or internal to Y . • Z is cohesive if all definable sets X and Y not orthogonal to Z are notorthogonal to each other. Proposition 3.2. (1) A definable set internal to a cohesive set is cohesive.(2) Any cohesive set is indecomposable.Proof. (1) follows from the observation that if A is not orthogonal to B , and B internal to C , then A is also not orthogonal to C .We prove (2). Let Z be a cohesive set and suppose that Z is internal to X × Y where X and Y are orthogonal. By definition there is m ∈ N and a surjective definable map f : X m × Y m → Z . Since X m and Y m are orthogonal, for the sake of our argumentwe can assume m = 1. So we have a surjective definable map f : X × Y → Z and we need to show that Z is X -internal or Y -internal. For x ∈ X and y ∈ Y ,let f x ( y ) = f ( x, y ) = f y ( x ). The image of f x is Y -internal and the image of f y is X -internal. These two images are then orthogonal (Proposition 2.12), so theycannot be both infinite by the hypothesis on Z . It follows that either Im( f x ) is finitefor all x ∈ X or Im( f y ) is finite for all y ∈ Y . By symmetry let us assume that ALESSANDRO BERARDUCCI, PANTELIS E. ELEFTHERIOU, AND MARCELLO MAMINO
Im( f y ) ⊆ Z is finite for all y . Let E y ⊆ X be the equivalence relation defined by xE y x ′ ⇐⇒ f ( x, y ) = f ( x ′ , y ). Then E y is a definable equivalence relation on X of finite index. Since ( E y ⊆ X × X | y ∈ Y ) is a definable family of subsets of an X -internal set indexed by a Y -internal set, by orthogonality there are finitely manysets of the form E y for y ∈ Y (Fact 2.4). The intersection E = T y ∈ Y E y is then againa definable equivalence relation of finite index on X . Let x , . . . , x k be representativesfor the equivalence classes of E . Then Z is the image of the restriction of f to S i ≤ k { x i } × Y , and therefore it is Y -internal. (cid:3) Proposition 3.3. If X and Y are cohesive and non-orthogonal, then X × Y is cohesive.Proof. Let A and B be non-orthogonal to X × Y . We need to prove that A and B are non-orthogonal. By Corollary 2.6 either X or Y is non-orthogonal to A . Similarly,either X or Y is non-orthogonal to B . There are four cases to consider, but bysymmetry we can consider the following two cases.Case 1. X is non-orthogonal to both A and B .Case 2. X is non-orthogonal to A , and Y is non-ortogonal to B .In the first case by the cohesiveness of X the sets A and B are non-orthogonal. Inthe second case, since X and Y are non-orthogonal and Y is non-orthogonal to B ,by the cohesiveness of Y we conclude that X is non-orthogonal to B , so we have areduction to the first case. (cid:3) We now turn to groups definable in o-minimal structures.
Theorem 3.4.
Let G be a definable group of dimension in an o-minimal structure M . Then G is cohesive.Proof. Let A and B be definable sets non-orthogonal to G . We need to prove that A and B are not orthogonal. Let us first concentrate on A . By Corollary 2.8 there are n ∈ N and a relation R ⊆ A n × G which is not a finite union of boxes. Let P ( G ) bethe power set of G and let f : A n → P ( G ) be defined by f ( a ) = { g ∈ G | ( a, g ) ∈ R } .Since f ( a ) has dimension ≤
1, its boundary δf ( a ), closure minus interior, is eitherempty or has dimension 0. By the assumption on R , the image of f is infinite, thusalso the image of δf : A n → P ( G ) is infinite, since in a group of dimension 1 there canonly be finitely many definable subsets with a given boundary. Now recall that a setof dimension zero is finite and a definable family of finite sets is uniformly finite. Sothere is k ∈ N such that δf ( a ) has at most k elements for every a ∈ A n . By orderingthe points of δf ( a ) lexicographically, we have a map h : A n → G k with infinite image.It follows that there is i ≤ k such that π i ◦ h : A n → G has infinite image, where π i : G k → G is the projection onto the i -th component. We have thus proved thatthere is a definable map f A : A n → G with infinite image. Similarly, there is l ∈ N and a definable map f B : B l → G with infinite image. Infinite definable subsets ofa one-dimensional group have non-empty interior, thus, compositing with a grouptranslation, we can assume that the two images have an infinite intersection. Therelation xQy : ⇐⇒ f A ( x ) = f B ( y ) then witnesses the non-orthogonality of A and B . (cid:3) Remark 3.5.
Notice that, by Theorem 3.4, any o-minimal expansion M of a group iscohesive, hence all sets definable in M are cohesive by 3.2. For Theorems 8.4 and 8.7it is, therfore, important to work in an arbitrary o-minimal structure. RTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS 9
4. Splitting and decompositionIn this section we work in an arbitrary structure M . Fix definable orthogonal sets X , . . . , X n . We introduce the notions of splitting and decomposition for functionsand groups, respectively, definable in the disjoint union of the sets X i . Definition 4.1.
The disjoint union F i X i of X , . . . , X n is the multi-sorted structurehaving X , . . . , X n as sorts and all ( X k , . . . , X k n n )-boxes as basic relations, where k , . . . , k n ∈ N . Remark 4.2.
Thanks to the orthogonality of X , . . . , X n , a definable set is definablein F i X i if and only if it is contained in some cartesian product Q kj =1 X t ( j ) with t ( j ) ∈{ , . . . , n } for all j = 1 , . . . , k . Notation. If S ⊆ Q kj =1 X t ( j ) we define π i : S → X k i i as the projection of S onto the X i -components, where k i is the number of indexes j with t ( j ) = i . For instance, if S ⊆ X × X × X , then π : S → X maps ( a, b, c )to ( a, c ).We now introduce the notion of splitting, which will be a crucial tool for our proofs. Definition 4.3.
Let f : A → B be a function definable in F i X i . We say that f splits (with regard to X , . . . , X n ) if for every x, y ∈ dom( f ) and every i ≤ n , π i ( x ) = π i ( y ) ⇒ π i ( f ( x )) = π i ( f ( y )) . That is, π i ( f ( x )) depends only on π i ( x ). Note that if f splits, then up to a permu-tation of the indexes we can write f = f × · · · × f n | A where f i : π i ( A ) → π i ( B ).We will be using the following two facts without specific mentioning. Fact 4.4.
Let f : A → B be a function definable in F i X i . Then there is a finitepartition D of dom( f ) into definable sets, such that for each D ∈ D , the restrictionof f to D splits.Proof. By orthogonality of X , . . . , X n , up to a permutation of the variables, f is afinite disjoint union S j f j , where f j = U ,j × · · · × U n,j and U i,j ⊆ π i ( A ) × π i ( B ). Weconclude observing that each f j splits. (cid:3) Splitting is preserved under composition in the following sense.
Fact 4.5.
Let h, f , . . . , f n be maps definable in F i X i , and suppose that the map h ◦ ( f × · · · × f n ) is defined (in the sense that the range of f × . . . × f n is containedin the domain of h ). If h, f , . . . , f n split, so does h ◦ ( f × · · · × f n ) .Proof. Straightforward. (cid:3)
We now turn to definable groups.
Definition 4.6.
A definable group G admits a decomposition (with respect to X , . . . , X n ) if there are definable subsets A , . . . , A n of G , such that G = A . . . A n ,and A i is internal to X i , for all i . Note that, since a set internal to a cohesive set is cohesive (Proposition 3.2(1)),a definable group admits an orthogonal decomposition (as in the introduction) if andonly if there are definable orthogonal cohesive sets X , . . . , X n , such that G admits adecomposition with respect to them. Observation 4.7.
Let H be a finite index subgroup of G . If H admits a decompositionwith respect to X , . . . , X n , then so does G .Our goal in the next sections is to prove that if G is definable in an o-minimalstructure and internal to X × . . . × X n , then it admits a decomposition in the senseabove. A relevant case is when G is contained in – as opposed to being internal to – X × . . . × X n . In this situation, by Remark 4.8 below, if the group operation of G splits, then G admits a decomposition. The converse, however, is not true (Example4.9). On the other hand, if G is a direct product of groups definable in X , . . . , X n ,respectively, then the group operation obviously splits. In the appendix, we will givean example in which the group operation splits, yet the group is not even definablyisomorphic to a direct product.The rest of this section contains some remarks that help to demonstrate the newlydefined notions. Remark 4.8.
Let (
G, µ, e ) be a definable group with G ⊆ X × . . . × X n . Then thefollowing are equivalent:(1) the group operation µ splits (with respect to X , . . . , X n );(2) there are definable groups H , . . . , H n , contained in X , . . . , X n , respectively,such that G < H × . . . × H n ;(3) there are definable groups H , . . . , H n , contained in X , . . . , X n , respectively,such that G is a finite index subgroup of H × . . . × H n ;and, if either of these holds, then G admits a decomposition. Proof.
Assume (1). Consider the projections H , . . . , H n of G on X , . . . , X n , respec-tively. The group operation µ , by the splitting hypothesis, takes the form µ (( x , . . . , x n ) , ( y , . . . , y n )) = ( µ ( x , y ) , . . . , µ n ( x n , y n ))It is straightforward to check that µ i is a group operation on H i , and (2) is thusestablished.We prove, now, that G has finite index in H × . . . × H n . Let π i : Q i X i → X i bethe projection onto the i -th component and let p i : Y j X j → Y j = i X j be the projection that omits the i -th coordinate. Fix an index i and consider anelement k of Q j = i H j . Let H i ( k ) = π i ( G ∩ p − i ( k )). Observe that L i := H i ( p i ( e )) isa subgroup of H i . We claim that L := L × . . . × L n has finite index in H × . . . × H n .Assuming the claim, since L < G , also G must have finite index. It suffices to provethat L i has finite index in H i . The coset hL i , for h ∈ H i , coincides with H i ( p i ( g )) forany g ∈ G ∩ π − i ( h ). Thus, in particular, the cosets of L i belong to the family H i ( − )indexed over Q j = i X j . This family is finite by Fact 2.4, and this concludes the proofof (3). RTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS 11
It is immediate that (2), hence also (3), implies (1). It remains to show that G admits a decomposition. To this aim, observe that L is decomposable: it is, in fact, theproduct of the subgroups G ∩ p − i ( p i ( e )) ∼ = L i . We conclude by Observation 4.7. (cid:3) Example 4.9.
We give an example of a group whose operation does not split, evenup to definable isomorphism, but the group admits a decomposition. Let R = R =( R , <, +) be two orthogonal copies of the real field structure, and fix a ∈ [0 , Z · (0 ,
1) + Z · (1 , a ), and define G = ([0 , , µ, µ = + mod Λ. Clearly, G = A + A , where A = [0 , × { } and A = { } × [0 , a = 0, then µ doesnot split. Moreover, a ∈ Q , if and only if G is definably isomorphic to a group whoseoperation splits, if and only if G is definably isomorphic to a direct product of groupsdefinable in R and R , respectively.5. Compact dominationLet again M be an arbitrary structure and G an abelian definable group. We calla definable set S ⊆ G generic if finitely many translates of it cover G (since weassume that G is abelian, we do not need to distinguish between left-generic andright-generic). We recall that NIP structures include both the o-minimal structures(e.g. the real field) and the stable structures (e.g. the complex field). We are mainlyinterested in o-minimal structures, but in this section we consider the larger NIP class.If M is a NIP structure and G is compactly dominated (see [16] for the definitions),then generic sets have a nice behaviour. Compact domination is a model-theoreticform of compactness: for instance a semialgebraic linear group G < GL ( n, R ) iscompactly dominated if and only if it is compact. The following proposition subsumesall we need about these notions. Proposition 5.1.
Let M be a NIP structure and G a compactly dominated abeliangroup definable in M . Then:(1) If the union of two definable subsets of G is generic, then one of the two isgeneric.(2) Suppose G = HK , where H and K are definable subgroups and K is normal.Then then S ⊆ G is generic if and only if it contains a set of the form AB where A is a generic subset of H and B is a generic subset of K . (3) If G and H are compactly dominated, then G × H is compactly dominated.Proof. Point (1) holds for all groups with finitely satisfiable generics ( fsg ) [9, Propo-sition 4.2], and compactly dominated NIP groups have fsg [16, Theorem 8.37], [17,Proposition 3.23], [16, Proposition 8.33].To prove point (2) we may assume M is κ saturated for some sufficiently bigcardinal κ . We make use of the infinitesimal subgroup G (see [16] for the definition).Specifically, we need to observe that in a compactly dominated group G , a subset S isgeneric if and only some translate of S contains G [1, Proposition 2.1]. If G = HK with H < G and K ⊳ G , then G = ( HK ) = H K [4, Theorem 4.2.5] (thisholds without the hypothesis that G is compactly dominated). Now suppose S ⊆ G is generic. Then a translate X = hS of S contains G . From the definition of Below we only need the case when G is the direct product of H and K . infinitesimal subgroup it follows that we can write H = T i ∈ I U i and K = T j ∈ J V j for some downward directed families of definable generic sets ( U i ) i ∈ I and ( V j ) j ∈ J with | I | < κ and | J | < κ . By κ -saturation \ ij ( U i V j ) ⊆ \ i U i · \ j V j = G (given g ∈ T ij U i V j , consider the type p ( x, y ) that says x ∈ T i U i , y ∈ T j V j and g = xy ). Again by compactness, there are i, j with U i V j ⊆ G . For this choice of i, j we have U i V j ⊆ X = hS . We can thus write S as the product of the generic sets A = h − U i and B = V j .Point (3) follows from [17, Corollary 3.17] the product of smooth measures issmooth) and the fact that a group is compactly dominated if and only if it has asmooth left-invariant measure [16, Theorem 8.37]. (cid:3) Fact 5.2.
Definably compact groups in an o-minimal structure are compactly domi-nated.Proof.
This was first proved for o-minimal expansions of a field in [8]. We give somebibliographical pointers to obtain the result for arbitrary o-minimal structures. Firstone shows that a definably compact group in an o-minimal structures has fsg [11,Theorem 8.6]. From this one deduces that G admits a generically stable left-invariantmeasure [16, Proposition 8.32]. In an o-minimal structure (and more generally ina distal structure), a generically stable measure is smooth [16, Proposition 9.26].Finally, a NIP group with a smooth measure is compactly dominated [16, Theorem8.41]. (cid:3) Proposition 5.3.
Let G be a compactly dominated group. Given two generic sets A ⊆ G and B ⊆ G , there is h ∈ G such that A ∩ hB is generic.Proof. There is a finite subset I ⊆ G such that A ⊆ S h ∈ I hB . By Proposition 5.1(1),there is h ∈ I such that A ∩ hB is generic. (cid:3)
6. Decomposition of compactly dominated abelian groups (NIP)In this section M is a NIP structure and X , . . . , X n are orthogonal definable sets. Theorem 6.1.
Let G be a compactly dominated abelian group contained in X × · · ·× X n . Then G admits a decomposition with respect to X , . . . , X n .Proof. Let P n : G n → G be the function sending ( x , . . . , x n ) to Q ni =1 x i . Since G is compactly dominated, so is G n (Proposition 5.1(3)). By Fact 4.4 and Proposition5.1(1), P n splits on a generic definable set S ⊆ G n . By Proposition 5.1(2) andinduction on n we find definable generic sets A , . . . , A n ⊆ G such that A × . . . × A n ⊆ S . By Proposition 5.3 (and induction on n ) we find a , . . . , a n ∈ G such that the set U = n \ i =1 a i A i is generic in G . Again by Fact 4.4 and induction, there is a generic set D ⊆ U suchthat for every i = 1 , . . . , n the function f i : x ∈ D a − i x RTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS 13 splits on D . Since D ⊆ a i A i , the image of f i is contained in A i . It follows that thefunction f × . . . × f n : D n → A × . . . × A n splits. Since P n splits on A × . . . × A n , we deduce that the function f = P n ◦ ( f × . . . × f n )splits on D n . Since G is abelian, f ( x , . . . , x n ) = a n Y i =1 x i (6.1)where a = Q ni =1 a i . By the orthogonality assumption, D is a finite union of sets of theform U × . . . × U n with U i ⊆ X i . By Proposition 5.1(1) one of these sets is generic,so by replacing D with a smaller set we can assume that D = U × . . . × U n . Let π i : Q i X i → X i be the projection onto the i -th component and let p i : Y j X j → Y j = i X j be the projection that omits the i -th coordinate. Now fix k ∈ D and let D i = p − i p i ( k ) ∩ D = { x ∈ D | ∀ j = i π j ( x ) = π j ( k ) } . Notice that D i is U i -internal, hence a fortiori it is X i -internal. We claim that k n − D = D . . . D n (6.2)To prove the claim, let g ∈ D and let x i ∈ G be such that π i ( x i ) = π i ( g ) & π j ( x i ) = π j ( k ) for j = i. Notice that x i ∈ D i . Since f splits on D n , the value of π i f ( x , . . . , x n ) does notchange if we replace x i with g and x j with k for j = i . By Equation (6.1) we thenobtain π i f ( x , . . . , x n ) = π i ( ak n − x ) . Since this holds for every i , we deduce that f ( x , . . . , x n ) = ak n − g and since f ( x , . . . , x n ) = a Q ni =1 x i we obtain Equation 6.2.We have thus shown that a translate of D is a product of X i -internal sets. Since D is generic, the same holds for G . (cid:3)
7. Decomposition of definably compact abelian groups (o-minimal)In this section M is an o-minimal structure and X , . . . , X n are orthogonal definablesets. We prove the following variant of Theorem 6.1, where the NIP hypothesis isreplaced by o-minimality, but the group is only assumed to be internal to X × . . . × X n . Theorem 7.1.
Let G be a definably compact abelian group internal to X × · · · × X n .Then G admits a decomposition with respect to X , . . . , X n . We need the following lemma.
Lemma 7.2.
Let X be an infinite set definable in an o-minimal structure M . Thenthere is a definable set Y = [ X ] o-min such that:(1) X and Y are internal to each other;(2) there is k ∈ N and a definable injective map from X to Y k ;(3) Y has a definable linear order ≺ such that ( Y, ≺ ) with the induced structurefrom M is o-minimal. We call the resulting structure Y the o-minimalenvelope of X .Proof. Suppose X ⊆ M m and let π i : M m → M be the projection onto the i -th coordinate ( i = 1 , . . . , m ). Fix parameters a < · · · < a m in M and let Z = S i { a i } × π i ( X ) ⊆ M . Then Z has dimension 1 and is bi-internal to X . Each π i ( X )is a finite union of open intervals and points with induced order from M . We order Z lexicographically, i.e. all the elements of { a i } × π i ( X ) preceed all the elements of { a i +1 }× π i +1 ( X ). Adding and removing from Z finitely many points we obtain a set Y satisfying point (3) (we need each pair of consecutive open intervals to be separatedby exactly one point). Points (1) and (2) are clear from the construction. (cid:3) Definition 7.3 ([7, Def. 1.1]) . Let
X, Y be definable sets, E , E two definableequivalence relations on X and Y respectively. A function f : X/E → Y /E is calleddefinable if the set { ( x, y ) ∈ X × Y | f ([ x ] E ) = [ y ] E } is definable. Proposition 7.4.
Let G be a definable group in an o-minimal structure M and let X , . . . , X n be definable sets in M . If G is internal to X × · · · × X n , then there isan injective definable map f : G → X ′ × · · · × X ′ n , where each X ′ i is bi-internal to X i ( i = 1 , . . . , n ).Proof. By Lemma 7.2 G is internal to Y = [ X × · · · × X n ] o-min , so it can be con-sidered as an interpretable group in the o-minimal structure Y . By [7] interpretablegroups in an o-minimal structure are definably (in the sense of Definition 7.3) isomor-phic to definable groups. It follows that there is a definable injective map f : G → Y k for some k ∈ N . By the construction of the o-minimal envelope, Y can be embeddedinto a product of sets Y ′ i , where Y i is bi-internal to X i (if X i ⊆ M n , it suffices to define Y i as π ( X i ) × · · · × π n ( X i )). Now it suffices to take X ′ i = Y ki . (cid:3) We are now ready to finish the proof of the theorem.
Proof of Theorem 7.1.
By Proposition 7.4 there are definable sets X ′ , . . . , X ′ n with X ′ i bi-internal to X i such that G is definably isomorphic to a group G ′ contained in X ′ × · · · × X ′ n . Since G is definably compact, G ′ also is. By Theorem 6.1 G ′ admitsa decomposition with respect to X ′ , . . . , X ′ n , hence also with respect to X , . . . , X n .Hence so does G . (cid:3)
8. Decomposition: general case (o-minimal)In this section, we prove our main theorems (8.4 and 8.7). Fix an o-minimal struc-ture M and definable orthogonal sets X , . . . , X n . We first prove that the existenceof decompositions is preserved under taking central extensions. The induced structure contains a predicate for each M -definable subset of Y n for n ∈ N . RTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS 15
Lemma 8.1.
Let → N → G f → H → be a definable exact sequence of definablegroups internal to X × · · · × X n with N < Z ( G ) . If N and H admits a decompositionwith respect to X , . . . , X n , then G too admits a decomposition with respect to thesame orthogonal sets.Proof. By assumption, we can write H = H . . . H n and N = N . . . N n , where H i and N i are X i -internal definable sets (not necessarily subgroups). We have G = f − ( H ) . . . f − ( H n ) . By [6, Theorem 2.5], there is a definable section σ : H → G . Since f − ( H i ) = σ ( H i ) N , we have G = σ ( H ) N . . . σ ( H n ) N. Since N = N . . . N n is contained in the center of G , it follows that G = U . . . U n ,where U i is the X i -internal set σ ( H i ) N i . . . N i ( n occurrences of N i ). (cid:3) We can now handle the abelian case.
Proposition 8.2.
Let G be a definable group internal to X × · · ·× X n . If G is abelian,then G admits a decomposition with respect to X , . . . , X n .Proof. We reason by induction on dimension. For dim( G ) = 1, G is cohesive byTheorem 3.4, so it is indecomposable (Proposition 3.2), hence it is internal to oneof the X i . Let dim( G ) >
1. If G is definably compact, then we conclude by 7.1.If G not, then by [12, Theorem 1.2], G has a 1-dimensional torsion-free definablesubgroup H < G . Since dim( H ) = 1, H admits a decomposition. By induction ondimension, so does G/H . Therefore, since G is abelian, we can apply Lemma 8.1. (cid:3) The following fact must be well-known, but we include a proof for completeness.
Fact 8.3.
Let G be a connected group definable in an o-minimal structure. If Z ( G ) is finite, then G/Z ( G ) is centerless.Proof. Let a ∈ G , such that aZ ( G ) is in the center of G/Z ( G ). We want to prove that a ∈ Z ( G ). We have that for all b ∈ G , a − b − ab ∈ Z ( G ). Since G is connected, theimage of the map f : G → Z ( G ) sending b ∈ G to a − b − ab ∈ Z ( G ) is connected.Since Z ( G ) is finite, f must be constant. Since f maps the identity e ∈ G to e , wehave a − b − ab = e . Thus a ∈ Z ( G ), as needed. (cid:3) We can now prove our first main result.
Theorem 8.4.
Let G be internal to X × · · · × X n where X , . . . , X n are definableorthogonal definable sets. Then G admits a decomposition with respect to X , . . . , X n .Proof. We observe that, since the connected component of the identity G has finiteindex in G , by Observation 4.7, it suffices to find a decomposition of G . We maythus assume that G is connected.We prove, now, the theorem, by induction on dim( G ). For dim( G ) = 0, it isobvious. Assume dim( G ) > Z ( G ) is finite. Then Z ( G ) admits a decomposi-tion, and by Lemma 8.1 it suffices to prove that H = G/Z ( G ) has a decomposition.Since Z ( G ) is finite, H is centerless (Fact 8.3). By [7], H is definably isomorphic to a definable group. By [13, Theorems 3.1 and3.2], it follows that there are definable real closed fields R , . . . , R k and definablelinear groups H i < GL ( n, R i ) such that H is definably isomorphic to H × · · · × H k . Adefinable real closed field in an o-minimal structure has dimension 1 [12, Theorem 4.1];thus, by Theorem 3.4 each R i is cohesive. It follows that the subgroups H , . . . , H k are themselves cohesive, hence each of them is internal to one of the X j . Therefore H admits a decomposition with respect to the orthogonal sets X , . . . , X n .Case 2. Suppose Z ( G ) is infinite. By the abelian case (Proposition 8.2), Z ( G ) hasa decomposition with respect to X , . . . , X n . By induction on the dimension, G/Z ( G )has a decomposition too. Therefore we can conclude by Lemma 8.1. (cid:3) As a by-product of the proof we obtain:
Proposition 8.5. If G is a definably simple, then G is cohesive.Proof. Let G be definably simple. By the proof of Theorem 8.4 there is a definable realclosed field R such that G is definably isomorphic to a definable subgroup of GL ( n, R ).Since dim( R ) = 1, by Theorem 3.4 R is cohesive. But GL ( n, R ) is internal to R ,thus all its definable subsets are cohesive. (cid:3) We now proceed towards our second main result.
Lemma 8.6.
Let G be an interpretable group. Then there are cohesive orthogonal -dimensional definable sets X , . . . , X n and an injective map h : G → Q ni =1 X i .Proof. By [7, Theorem 3], there is a definable injective map f : G → Q kj =1 G j whereeach G j is a 1-dimensional definable group. Define a relation R on { , . . . , k } by iRj if G i and G j are not orthogonal. By Theorem 3.4, the groups G i are cohesive, hence R is an equivalence relation. Suppose there are n equivalence classes. Let X i be theproduct of the groups G j , with j in the i -th class. Using f , it is easy to define (by apermutation of the coordinates on the image) the injective map h : G → Q ni =1 X i .The sets X i are cohesive by Proposition 3.3 and mutually orthogonal by Corol-lary 2.6. (cid:3) Theorem 8.7. If G is a group interpretable in an o-minimal structure M , then G admits an orthogonal decomposition G = A . . . A n .Proof. Let X , . . . , X n be the sets provided by Lemma 8.6. By Theorem 8.4, thereare X i -internal sets A i ⊆ G , i = 1 , . . . , n , such that G = A . . . A n . Clearly, the A i ’s are orthogonal and cohesive, as they inherit those properties fromthe X i ’s. (cid:3) Corollary 8.8. If G is infinite, in Theorem 8.7 we can choose each A i to be infinite.In this case, the number n is an invariant of G up to definable isomorphism. Indeed,if G = B . . . B m is another decomposition of G as a product of orthogonal cohesiveinfinite definable subsets, then m = n and each B i is bi-internal to a unique A j .Proof. Suppose that G is infinite and fix an orthogonal decomposition G = A . . . A n .Then at least one A i is infinite, say A . If some A i is finite we may replace A with A A i and omit A i obtaining another valid orthogonal decomposition. So we may assume RTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS 17 that A , . . . , A n are all infinite. Now consider another decomposition G = B . . . B m into infinite orthogonal cohesive sets. Fix B i and observe that B i is internal to G ,which is internal to the cartesian product A × . . . × A n . Since B i is indecomposable,it must be internal to some A j . Moreover, j must be unique, because if B i is internalto both A j and A h , with j = h , then it is non-orthogonal to both, so by cohesivenessof B i , A i and A h are non-orthogonal, a contradiction. The argument also shows that m = n and B i is in fact bi-internal to the corresponding A j . (cid:3)
9. Locally definable groupsIn this section, we fix again an o-minimal structure M , and prove Theorem 9.3.Let us first recall a few definitions concerning locally definable sets and groups (whichcan, in fact, be given for arbitrary structures). Definition 9.1. A locally definable set X is a countable union of definable setstogether with a given presentation as such a countable union. A subset of a locallydefinable set X is said to be definable if it definable in M and is contained in theunion of finitely many sets of the presentation of X (this last condition is automaticallysatisfied if M is ℵ -saturated). A compatible subset of a locally definable set X is asubset which intersects every definable subset of X at a definable set. A compatiblesubset is discrete if it intersects every definable set into a finite set. A locallydefinable function is a function between locally definable sets whose restriction toeach definable set is definable. Similar definitions apply to groups. A locally definablegroup is a locally definable set with a locally definable group operation. We can thenspeak of compatible and discrete subgroups and locally definable homomorphisms. Alocally definable group is definably generated if it is generated by a definable subset. Definition 9.2.
Given finitely many structures X , . . . , X n , their disjoint union F i X i is the multi-sorted structure with a sort for each X i and whose basic relations are thedefinable sets in the single structures X i .Notice that if X i is the domain of X i , then X , . . . , X n are orthogonal in F i X i . Theorem 9.3.
Let G be an abelian group definable in the disjoint union F i X i of finitelymany o-minimal structures X , . . . , X n . Then there is a locally definable isomorphism G ∼ = G × . . . × G n / Γ , where G i is a locally definable and definably generated group in X i , and Γ is a com-patible locally definable discrete subgroup of G × . . . × G n .Proof. The structure F i X i is bi-interpretable with the o-minimal structure M ob-tained by concatenating X , . . . , X n in the given order and adding n − X i from X i +1 for i < n . We can therefore apply to M = F i X i the variousresults concerning o-minimal structures. By Theorem 8.4, the group G admits a de-composition G = A . . . A n with respect to X , . . . , X n , where X i is the domain of X i .Let h A i i be the locally definable subgroup of G generated by A i .We claim that h A i i is locally definably isomorphic to a definably generated group G i in the structure X i . To this aim, let A ( n ) i ⊆ G consist of the n -fold products a . . . a n ,where each a i is either an element of A i or is the group-inverse of an element of A i . Choose k , . . . , k n ∈ N such that G is included in X k × . . . × X k n n . Since A ( n ) i is X i -internal and included in G , it must have a finite projection on the factors differentfrom X i . Thus we can write A ( n ) i ⊆ X k i i × F n where F n is a finite set. It followsthat the subgroup h A i i of G generated by A i is included in X k i i × F where F = S n F n is a countable set. Consider a bijection sending F to a countable subset of X i .This induces a locally definable bijection between h A i i and a locally definable subset G i ⊆ X k i +1 i . We can endow G i with a group operation via the bijection. The resultinggroup G i will then be locally definable, and in fact definably generated, in the structure X i . There is a locally definable group homomorphism f : G × . . . × G n → G inducedby the composition G × . . . G n ∼ = h A i × . . . × h A n i → G , where the last map sends( x , . . . , x n ) to their product in G . Note that f is a homomorphism since G is abelian.It remains to prove that the kernel Γ of the above f is discrete. To this aim fix,for i = 1 , . . . , n , a definable subset U i of h A i i and let S be the set of all tuples( a , . . . , a n ) ∈ U × . . . × U n such that a . . . a n = 1 G . It suffices to show that S isfinite. The sets U , . . . , U n are orthogonal by Proposition 2.12, since they are internalto A , . . . , A n respectively. It follows that S is a finite union of sets of the form B × · · · × B n with B i ⊆ U i . However, each B i can only be a singleton because anychoice of n − a , . . . , a n determines the last one via the equation a . . . a n = 1 G . (cid:3)
10. Questions(1) Does the conclusion of Theorem 8.4 extend to the case when M is an arbi-trary structure?(2) Given a structure M and a definably simple group G in M , is G alwayscohesive? (The answer is positive if M is o-minimal, by Proposition 8.5 Onecould then consider the case when M is merely assumed to be NIP.)(3) Can the abelianity hypothesis in Theorem 9.3 be removed?(4) Are groups of dimension 1 in a geometric theory always cohesive? (Theanswer is positive in the o-minimal context, by Theorem 3.4.)(5) Are indecomposable sets always cohesive?(6) Is a set internal to an indecomposable set also indecomposable?(7) Does Proposition 2.9 hold without the saturation hypothesis?(8) Is it true that if a definable group G admits a decomposition with regard toorthogonal definable sets X , . . . , X n , then so does every definable subgroupof G ? 11. AppendixWe construct an example of a definable group whose group operation splits, butthe group is not a product of two infinite groups, hence in particular two orthogonalgroups. First we need the following observation. Example 11.1.
There is a real Lie group G , definable in the pure real field structure,which is not a semidirect product of the connected component of the identity G anda finite non-trivial group. Proof.
Let p be an odd prime. The Heisenberg group mod p is the semialgebraicgroup H of matrices of the form a c b where c ∈ R /p Z and a, b ∈ Z /p Z . We RTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS 19 claim that H is not Lie isomorphic to a semidirect product of a connected real Liegroup and a discrete group. Taking a, b = 0 we obtain the center Z ( H ) of H , whichcoincides with H and it is isomorphic to the circle group R /p Z . The quotient H/H is isomorphic to ( Z /p Z ) . Since H is not abelian, H is not isomorphic to the directproduct H × ( Z /p Z ) . Moreover the direct product is the only possible semidirectproduct in the Lie category because ( Z /p Z ) has no non-trivial continuous action on R /p Z (since the only non-trivial definable automorphism of R /p Z is the inverse, and p is odd). (cid:3) Example 11.2.
Let R and R be two orthogonal copies of the field R and workin the o-minimal structure M = R ⊔ R obtained by concatenation of R and R with a separating element between them. Let H i be the Heisenberg group mod 3over R i . Consider the definable group H × H . Now consider the definable subgroup G < H × H consising of the pairs of matrices * a c b , a c ′ b + with a, b ∈ Z / Z , c, c ′ ∈ R / Z . Note that G is definably isomophic to an extensionof ( R / Z ) by ( Z / Z ) . In Proposition 11.3 below we prove that G is not a directproduct of two infinite definable subgroups. This shows that, although the groupoperation splits with respect to R and R (because it is induced by the direct product H × H ), G is not a direct product of orthogonal subgroups. Proposition 11.3.
The group G in Example 11.2 is not a direct product of two infinitedefinable subgroups.Proof. Assume that G is the direct product of two definable infinite subgroups G and G . Then dim( G ) = dim( G ) = 1. We may assume that G is a R -internal and G is R -internal. Consider the natural (surjective) projections π i : G → H i .We claim that G = π − (0) and G = π − (0) where G i is the connected com-ponent of the identity of G i . Consider for instance G . Clearly π ( G ) is finite byorthogonality. Hence π − (0) ∩ G has finite index in G , so it is infinite. Moreover π − (0) is H × { } , hence it is connected and, since two definably connected one di-mensional groups having infinite intersection coincide, it must coincide with G . Theclaim is thus proved.Observe that Z ( G ) = G = G × G , thus [ G : G ][ G : G ] = [ G : G ] =9. So, there are three cases for the possible values of the indexes of G and G :(1 , , (3 , , (9 , G : G ] = 1 and [ G : G ] = 9 (observe that the third case issymmetric). In this case G is connected, thus G = G , and | π ( G ) | = 9 because G has index 9 in G = π − (0). On the other hand π ( G ) = π ( G ) π ( G ), that is H × { } = Z ( H × { } ) π ( G ). Since Z ( H ) ∼ = Z ( H × { } ) has index 9 in H , wehave H ∼ = Z ( H ) × π ( G ), contradicting the claim in Example 11.1.Second case: [ G : G ] = 3 = [ G : G ]. In this case we show that G and G areabelian and we reach a contradiction since G is not abelian. By symmetry it sufficesto show that G is abelian. First recall that G < Z ( G ), so in particular G is centralin G , and definably isomorphic to R / Z . It follows that G is definably isomorphic to a central extension of R / Z by Z / Z . We claim that such a group is necessarilyabelian. To this aim we first show that there is a copy of Z / Z which is a complementof G . Let G , aG , bG be the three connected components of G . Note that themap x x has image contained in G and its restriction to G is onto. Considerthe map sending x ∈ G to ( ax ) ∈ G . Since G is central, ( ax ) = a x . Nowlet y ∈ G be such that y = ( ax ) . Then axy − has order three and generates acomplement of G . We have thus proved the claim and we can conclude that G is a definably isomorphic to a semidirect product of R / Z by Z / Z . Finally observethat Z / Z does not admit continuous non trivial actions on R / Z , so the semidirectproduct must be direct and G is abelian. (cid:3) References[1] Alessandro Berarducci. Cohomology of groups in o-minimal structures: acyclicityof the infinitesimal subgroup.
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Universit`a di Pisa, Dipartimento di Matematica, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy
Email address : [email protected] Department of Mathematics and Statistics, University of Konstanz, Box 216, 78457 Konstanz,Germany; ANDDepartment of Mathematics, University of Pisa, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy.
Email address : [email protected] Universit`a di Pisa, Dipartimento di Matematica, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy
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