GGROUPS DEFINABLE IN TWO ORTHOGONAL SORTS
ALESSANDRO BERARDUCCI AND MARCELLO MAMINO
Abstract.
This work can be thought as a contribution to the model theoryof group extensions. We study the groups G which are interpretable in thedisjoint union of two structures (seen as a two-sorted structure). We showthat if one of the two structures is superstable of finite Lascar rank and theLascar rank is definable, then G is an extension of a group internal to the(possibly) unstable sort by a definable subgroup internal to the stable sort. Inthe final part of the paper we show that if the unstable sort is an o-minimalexpansion of the reals, then G has a natural Lie structure and the extensionis a topological cover. Introduction
This paper can be thought as a contribution to the model-theory of covers ofgroups in the spirit of [17, 8, 3]. We assume some familiarity with the basic notionsof model theory, but we recall some relevant definitions in §
2. A good recentreference is [15].Given two structures Z and R , let ( Z, R ) be the two-sorted structure with a sortfor Z and another sort for R in a disjoint language (no connections between thetwo sorts). Note that Z and R are then fully orthogonal in the following sense: anydefinable subset of Z m × R n is a finite union of sets of the form A × B with A adefinable subset of Z m and B ad definable subset of R n .Our aim is to study the groups G which are interpretable in ( Z, R ), or equiva-lently definable in (
Z, R ) eq (see § H × K with H definable in Z and K definable in R .More generally one can have a quotient of H × K by a finite subgroup. There arehowever more interesting examples like the following. Example 1.1. ([8]) The universal cover f : G → H of a real Lie group H definablein an o-minimal expansion R of the real field is interpretable in (( Z , +) , R ).A few comments are in order. By [5] the universal cover f : G → H of H can berealized as a locally definable group and admits a definable section s : H → G (seealso [2]). In [8, §
8] it is showed that the bijection G → ker( f ) × H induced by thesection gives an intepretation of ( G, f, R ) in the two-sorted structure (ker( f ) , R ).On the other hand since ker( f ) ∼ = π ( H ) is abelian and finitely generated, ker( f )is interpretable in ( Z , +) and therefore f : G → H is interpretable in the two-sorted structure (( Z , +) , R ). In the same way one shows that any cover of H isinterpretable in (( Z , +) , R ). (See also [3, Prop. 3.1].) Date : Date: 13 March 2013.2010
Mathematics Subject Classification.
Key words and phrases.
Definable groups, stability, o-minimality.Partially supported by PRIN 2009WY32E8 003 “O-minimalit`a, teoria degli insiemi, metodi emodelli non standard e applicazioni”.Partially supported by Funda¸c˜ao para a Ciˆencia e a Tecnologia grant SFRH/BPD/73859/2010.We also acknowledge the support of FIRB2010, “New advances in the Model Theory ofexponentiation”. a r X i v : . [ m a t h . L O ] A p r A. BERARDUCCI AND M. MAMINO
Example 1.1 shows that a group G interpretable in ( Z, R ) eq does not need toarise from a direct product. The next natural question is whether G is always anextension of a group definable in one sort by a group definable in the other sort.We will show that this is indeed the case under a suitable stability assumption on Z , but let us first show that in full generality the question has a negative answer.To this aim we take Z = R = ( R , + , < ). So we have two structures Z and R whichare at the same time “equal” and orthogonal. There is of course no contradiction:indeed strictly speaking in ( Z, R ) we only have an isomorphic copy of Z and anisomorphic copy of R with the isomorphism not definable in ( Z, R ). Example 1.2.
Let Z = R = ( R , + , < ). There is a group G definable in ( Z, R ) eq with no infinite definable subgroup internal to one of the two sorts. So in particular G cannot be a definable extension of a group internal to one sort by a group internalto the other sort. Proof. (Based on [11, Example 5.2]) Take G = ( Z × R ) / Λ with Z ∼ = Λ < Z × R and Λ in sufficiently generic position. Note that Λ is not definable. However we candefine G in ( Z, R ) eq taking a definable set X ⊆ Z × R such that X + Λ = G and X ∩ Λ is finite (a big enough square X = [0 , a ] × [0 , a ] will do) and identifying G with X/ Γ (where X/ Λ is the quotient of X by the equivalent relation “to be in thesame coset”). Since we only need a finite portion of Λ to define X/ Λ we obtain adefinition in (
Z, R ) eq . This is exactly the example in [11] except that in that paperthe authors work with only one sort (which amounts to have the identity map from Z to R at disposal). They prove that in the one-sort setting G has no definableproper infinite subgroups. This holds a fortiori in the two-sorted setting since wehave fewer definable sets. Thus clearly G has no infinite subgroups internal to oneof the two sorts. (cid:3) In Example 1.2 it is important to have the order relation < in the language,so the structures are unstable (in the model-theoretic sense). If we work with thestable structures Z = R = ( R , +) the argument breaks down since in this case weare not able to define G = ( Z × R ) / Λ in (
Z, R ) eq . We will show that under asuitable stability assumption on the Z -sort any group interpretable in ( Z, R ) is anextension of a group interpretable in R by a group interpretable in Z . Let us recallthat in a stable theory the SU-rank coincides the U -rank or “Lascar rank” [15, 9].Our main result is: Theorem (See 7.1) . Let Z be a superstable structure of finite SU -rank and assumethat the SU -rank is definable. Let R be an arbitrary structure. Given a group ( G, · ) definable in ( Z, R ) eq , there is a Z -internal definable normal subgroup Γ (cid:67) G suchthat G/ Γ is R -internal. Note that in any superstable structure Z of SU-rank 1 (for instance ( C , + , · ) or( Z , +), or ( R , +)), the SU-rank is definable (see [13]), and therefore Z satisfies theassumption of the theorem.The subgroup Γ will in general depend on how G sits in the ambient space( Z, R ) eq and not only on the definable isomorphism type of G . In particular Γ isneither the minimal nor the maximal Z -internal normal subgroup such that G/ Γ is R -internal. For instance if G is the universal cover of the circle group R / Z , then G can be naturally interpreted in (( Z , +) , R ) by [8], but it has no minimal or maximal Z -internal normal subgroup. In this example Γ is Z -internal if and only if 2Γ issuch, and G/ Γ is R -internal if and only if G/
2Γ is such, so there is no reason toprefer Γ over 2Γ.The subgroup Γ is easier to describe if G is definable in ( Z, R ) rather than(
Z, R ) eq . In this case we have G ⊆ Z m × R n for some m, n ∈ N and we can consider the projection π R : Z m × R n → R n . We then define:Γ = (cid:8) g ∈ G : (Most y )(Most x ) (cid:0) π R ( xg y ) = π R ( g y x ) = π R ( x ) (cid:1)(cid:9) where g y = ygy − and (Most y ) φ ( y ) means that the projection on Z m of the set of y ∈ G such that φ ( y ) fails has lower SU-rank than the projection of the whole of G . The definition of Γ when G is definable in ( Z, R ) eq is similar, but we need toredefine π R to give meaning to the formula. To do this we will first show thatthere is a finite-to-one function f from ( Z, R ) eq to Z eq × R eq (uniform in eachsort). Composing with the projection from Z eq × R eq to R eq we obtain the desiredsubstitute for π R and the same definition of Γ will then work.In § Theorem (See 9.7) . If Z is an arbitrary structure and R is o-minimal, then everygroup G definable in ( Z, R ) admits a unique “ t -topology” in analogy with the o-minimal case (treated in [12] ). In particular, if R is based on the reals, then G has a natural Lie group structure.Combining Theorem 9.7 and Theorem 7.1 we then obtain: Corollary (See 9.14) . If R is o-minimal and Z is superstable of finite Lascar rank,any group definable in ( Z, R ) is a cover of a group definable in R . Here by “cover” we mean a definable morphism which is continuous and open inthe t-topology and has a discrete kernel. Note that there are extensions of groupsdefinable in R = ( R , + , · ) by ( Z , +) that are not covers (see [3, Theorem 3.12]).Among all the extensions, only the covers will be definable in (( Z , +) , R ) eq . Thefollowing example may be instructive: Example 1.3. ([3, Theorem 3.12]) There is an exact sequences 0 → ( Z , +) → G f → H → G ∼ = ( R , +), H ∼ = R / Z , and f (1 /n ) ∈ H does not convergefor n → ∞ (where 1 is any fixed element of G ). So in particular we cannot puta compatible topology on G making f into a covering. (Hence the morphism f : G → H is not interpretable in (( Z , +) , R )).The paper is organized as follows. In § Z, R ) eq . In § Z, R ) eq to Z eq × R eq and estab-lish its properties. In § Z is superstable of finite SU-rank and wedefine the dimension of a type in ( Z, R ) eq as the SU-rank of its projection to Z eq (using the projections studied in § Z eq we derive a similar results for ( Z, R ) eq (Proposition 4.4). We will sometimeconsider types over “unbounded” sets of parameters, namely sets of parameterswhose size depends on the model. In particular in Corollary 4.5 we have a set ofparameters including the whole of R and we are nevertheless able to find a suitablerealization of the type (this will play a crucial role in the proof of the main theo-rem). In § Z, R ) eq as the maximaldimension of the types of its elements. We can equivalently define the dimensionof a definable set in ( Z, R ) eq as the SU-rank of its “projection” to Z eq . However inorder to prove the invariance of the dimension under definable bijections it is moreconvenient to use the approach via types (since projections do not commute withbijections). In most of the lemmas we do not really need the stability assumptionbut only the additivity of the SU-rank (which holds also in supersimple theories offinite SU-rank). However superstability is used in Proposition 5.2 and to prove thedensity property in Theorem 5.4. In § x ” A. BERARDUCCI AND M. MAMINO based on the dimension on (
Z, R ) eq . This gives us a convenient notation to definethe subgroup Γ whose existence is asserted in the main theorem. It is important toassume that in Z the SU-rank is definable in order to ensure that the quantifier “formost x ” is a definable operation. Finally in § § § R is o-minimal. Acknowledgements.
The results of this paper were obtained through a longchain of successive generalizations in the course of which the hypothesis were pro-gressively weakened. Part of this process was stimulated by conversations withAnand Pillay on the occasion of the meetings “Model Theory in Algebra, Analysisand Arithmetic” (Cetraro, 2-6 July 2012) and “Model Theory: Groups, Geometry,and Combinatorics” (Oberwolfach 6-12 Jan. 2013). We also thank Ya’acov Peterzilfor his suggestion to study the subgroup Γ · C G (Γ) (see Corollary 8.3). Prelimi-nary versions of the results were presented at the “Konstanz-Naples Model TheoryDays” (Konstanz 6-8 Dec. 2012) and at the Oberwolfach meeting.2. Orthogonality and internal sets
Given a complete first order theory T we usually denote by C the monster modelof T (see [15]). We can think of C as a proper class model which is κ -saturated forevery cardinal κ . Every small (i.e. set-sized) models of T can be embedded in C as an elementary substructure, so we can assume that all the models that we areinterested are elementary substructures of C .Two definable subsets Z and R of C are fully orthogonal if every definable subset X of Z × R is a finite union (cid:83) i ∈ I U i × V i of “rectangles” U i × V i with U i a definablesubset of X and V i a definable subset of R . It can be shown that if Z and R arefully orthogonal, then so are Z m and R n , but to avoid the verification we can aswell include this condition in the definition of full orthogonality.A definable set X in C is stably embedded if every subset of X definable withparameters from C is definable with parameters from X . In this case we consider X as a structure on its own right with a symbol for each ∅ -definable subset of X n for any n .Given two definable sets X and V in C , X is said to be internal to V , or V -internal , if X is in the definable closure of V and a finite set of parameters. Equiva-lently there is a definable surjection from V n to X for some n (see for instance [15,Lemma 10.1.4]). We recall that C has elimination of imaginaries if, by definition,every definable set X in C has a code, where a code for X is a finite tuple c ofelements of C such that for every automorphism f of C we have that f fixes c if andonly if f fixes X setwise. All the codes for the same set X are interdefinable and X is definable over any of its codes. We follow the common convention of denotingby (cid:112) X (cid:113) a code for X .As usual C eq denotes the expansion of C with imaginary sorts: for each n ∈ N and each ∅ -definable equivalence relation E on C n we have a sort S E interpretedas C n /E together with the natural projection π E : C n → C n /E . Given a ∈ C n the E -equivalence class of a can be seen in two ways: as a definable subset [ a ] E of C n ,or as an element a/E of S E . The structure C eq has elimination of imaginaries. Inparticular the definable set [ a ] E can be coded by the element a/E ∈ S E .The advantage of working in C eq is that quotients of definable sets by definableequivalence relations become definable. It follows in particular that a group isinterpretable in C if and only if it is isomorphic to a group definable in C eq . Assumption 2.1.
In the rest of the paper we assume that Z and R are stablyembedded and fully orthogonal definable subsets of a monster model C . The subsetsof Z m × R n definable in C are exactly (up to a natural identification) the definablesets in the two sorted structure ( Z, R ) (with no connections between the two sorts).
Unless otherwise stated in the sequel by “definable” we mean definable in (
Z, R ) eq (possibly with parameters from the monster model).By symmetry all the results depending only on Assumption 2.1 hold with theroles of Z and R interchanged. This applies in particular to the following: Lemma 2.2.
Let ( X t ) t ∈ Z m be a definable family of subsets of R n indexed by Z m .Then { X t : t ∈ Z m } is finite. More generally the same holds for a definable family ( X t ) t ∈ Y of subsets of an R -internal set X indexed by a Z -internal set Y (where X and Y are definable sets in ( Z, R ) eq ).Proof. For the first part consider the definable set X = { ( t, x ) : x ∈ X t } ⊆ Z m × R n .By full orthogonality we can write it as a finite union of definable sets of the form A × B with A ⊆ R n , B ⊆ Z m and the desired result follows at once. For the secondpart let f : R n → X and g : Z n → Y be definable surjective maps witnessinginternality to the respective sorts. By the first part { f − ( X g ( z ) ) : z ∈ Z m } is finite.So { X t : t ∈ Y } is also finite. (cid:3) Proposition 2.3.
For a definable set X ⊆ Z m × R n the following are equivalent:(1) X is Z -internal.(2) The projection of X on the R -coordinates is finite.(3) There is a definable bijection from X to a definable subset of Z m +1 .Proof. (1) implies (2): Suppose X is Z -internal. Then there is a definable surjectivemap f : Z k → X for some k . Composing with the projection π R : Z m × R n → R n we obtain a map f from Z k to R k whose image is finite by Lemma 2.2. On the otherhand the image of f coincides with the projection of X onto the R -coordinates.(2) implies (3): Suppose X ⊆ Z m × F where F ⊆ R n is finite. Fix a bijection f from F to a finite subset of Z . Then f induces a bijection from X to a definablesubset of Z m +1 .(3) implies (1): Assume (3). Then clearly there is a definable surjective mapfrom Z m +1 to X , so X is Z -internal. (cid:3) Lemma 2.4.
Let X be a definable set in ( Z, R ) eq which is both Z -internal and R -internal. Then X is finite.Proof. By the hypothesis there are m, n ∈ N and definable surjective functions f : Z m → X and g : R n → X . For x ∈ Z m let H ( x ) = g − ( f ( x )) ⊆ R n . ByLemma 2.2 the family of sets { H ( x ) : x ∈ Z m } is finite. Since distinct elements of X have disjoint preimages through g , it follows that X is finite. (cid:3) Let us recall that, given a set A of parameters in some structure, the definableclosure dcl( A ) is the set of points which are definable over A and the algebraicclosure acl( A ) is the set of points which belong to some finite set definable over A .Clearly acl( A ) ⊇ dcl( A ). Lemma 2.5.
Let A be a set of parameters from ( Z, R ) eq . We have:(1) Let X ⊆ Z m × R n be definable over A . Then we can write X = (cid:83) ki =1 U i × V i with U i ⊆ Z m definable over acl( A ) ∩ Z eq and B i ⊆ R m definable over acl( A ) ∩ R eq . In particular X is definable over (acl( A ) ∩ Z eq ) ∪ (acl( A ) ∩ R eq ) .(2) Let a be an element of ( Z, R ) eq . Then a is definable over (acl( a ) ∩ Z eq ) ∪ (acl( a ) ∩ R eq ) .(3) Let X be a definable subset of some sort of Z eq and suppose that X isdefinable over A . Then X is definable over dcl( A ) ∩ Z eq .(4) For every set of parameters A from ( Z, R ) eq we have acl( A ∪ R ) ∩ Z eq =acl( A ) ∩ Z eq . A. BERARDUCCI AND M. MAMINO (5) The type of an element of Z eq over A ∪ R , is implied by its type over acl( A ) ∩ Z eq .(6) Given a tuple b from ( Z, R ) eq , tp( b/R ) is implied by tp( b/ dcl( b ) ∩ R eq ) .More generally tp( b/A ∪ R ) is implied by tp( b/A ∪ (dcl( b ) ∩ R eq )) . Note that in (5) and (6) a type over the big set of parameters R = R ( C ) isimplied by a type over a small set of parameters. Proof. (1) By considering the sections X r = { x ∈ Z n : ( x, r ) ∈ X } with r ∈ R m we obtain a finite boolean algebra of definable subsets of Z n . The atoms of thisboolean algebra are permuted by any automorphism of the monster model fixing X setwise, so they can be coded by elements in acl( A ) ∩ Z eq (the codes can be takenin Z eq by stable embeddedness). It follows that each set in the boolean algebra isdefinable over acl( A ) ∩ Z eq . Similarly, considering the sections over Z n , we obtaina finite boolean algebra of subsets of R m . If U ⊆ Z m and V ⊆ R n are atoms of therespective boolean algebras, then U is definable over acl( A ) ∩ Z eq , V is definableover acl( A ) ∩ R eq and U × V is either contained or disjoint from X . The desiredresult follows.(2) Let X ⊆ Z m × R n be the equivalence class corresponding to a ∈ ( Z, R ) eq and apply (1).(3) By stable embeddedness X is definable with parameters from Z eq so it hasa code (cid:112) X (cid:113) in Z eq . On the other hand since X is definable over A , we have (cid:112) X (cid:113) ∈ dcl( A ). So X is definable over dcl( A ) ∩ Z eq .(4) Without loss of generality we can assume A = ∅ . Let b ∈ acl( R ) ∩ Z eq . Thenthere is a tuple r from R and an algebraic formula φ ( x, r ) such that φ ( b, r ) holds.Let N ∈ N be the cardinality of X r = { x : φ ( x, r ) } . We can assume that for alltuples r (cid:48) from R of the same length as r , the set X r (cid:48) = { x : φ ( x, r (cid:48) ) } has cardinalityat most N . Each X r (cid:48) is a subset of a given sort of Z eq (the sort of b ), and thefamily of these sets is indexed by R k for some k (the length of the tuple r ). ByLemma 2.2 it follows that the family of sets { X r (cid:48) } r (cid:48) ∈ R k is finite, and since each ofthem is finite, the union (cid:83) r (cid:48) X r (cid:48) is finite. Now it suffices to observe that this unionis ∅ -definable and contains b .(5) We can assume A = ∅ . Let e ∈ Z eq and let φ ( x, r ) ∈ tp( b/R ). By point(3) (with A = { r } ) φ ( x, r ) is equivalent to a formula ψ ( x ) with parameters fromdcl( r ) ∩ Z eq ⊆ acl( ∅ ) ∩ Z eq , where the inclusion follows from point (4).(6) It suffices to prove the case A = ∅ . Let φ ( x, r ) ∈ tp( b/R ) where r is a tuplefrom R , say r ∈ R k . So r ∈ Y := { y ∈ R k : φ ( b, y ) } . By stable embeddedness Y canbe defined with parameters from R , so it has a code (cid:112) Y (cid:113) in R eq . On the other handwe must also have (cid:112) Y (cid:113) ∈ dcl( b ), so φ ( b, y ) is equivalent to a formula ψ ( y ) definableover dcl( b ) ∩ R eq . To conclude it suffices to observe that the formula ∀ y ( ψ ( y ) → φ ( x, y )) belongs to tp( b/ dcl( b ) ∩ R eq ) and implies φ ( x, r ) (take y = r ). (cid:3) Imaginaries
In this section we define a finite-to-one function from (
Z, R ) eq to Z eq × R eq andstudy its properties. Lemma 3.1.
Given a definable family ( X t ) t ∈ Y of definable susets of Z m indexedby a definable set Y in ( Z, R ) eq , there is a uniform family ( (cid:112) X t (cid:113) ) t ∈ Y of codes inthe following sense:(1) For each t ∈ Y the set X t ⊆ Z m is coded by (cid:112) X t (cid:113) ∈ Z eq and the function t (cid:55)→ (cid:112) X t (cid:113) from Y to the appropriate sort of Z eq is definable.(2) For all t, t (cid:48) ∈ Y we have (cid:112) X t (cid:113) = (cid:112) X t (cid:48) (cid:113) if and only if X t = X t (cid:48) . Proof.
Given t ∈ Y , by stable embeddedness there is a formula ϕ ( − , b ) with param-eters b from Z which defines X t (where “ − ” is the free variable of the formula). Bycompactness there is a finite collection of formulas φ , . . . , φ k such that for every t ∈ Y there is i ≤ k and a tuple b from Z such that X t is defined by φ i ( − , b ).Let bE i b (cid:48) if and only if φ i ( − , b ) is equivalent to φ i ( − , b (cid:48) ) and let a = b/E i ∈ Z eq be the corresponding imaginary element. Define f ( t ) = ( i, b/E i ), where X t is de-fined by φ i ( − , b ) and i is minimal such that there exists such a tuple b . Identifyingthe indexes 1 , . . . , k with tuples from Z we can consider f ( t ) as an element of theappropriate sort of Z eq and define (cid:112) X t (cid:113) = f ( t ). (cid:3) Definition 3.2.
Let S E = ( Z m × R n ) /E be a sort of ( Z, R ) eq . For a ∈ S E let π − E ( a ) ⊆ Z m × R n be the equivalence class corresponding a and let π Z ( π − E ( a )) ⊆ Z m be its projection on Z m . For a ∈ S E define a Z = (cid:112) π Z ( π − E ( a )) (cid:113) ∈ Z eq where the codes are chosen uniformly as in Lemma 3.1 (so a (cid:55)→ a Z is definable).Note that when a ∈ Z m × R n , then a (cid:55)→ a Z ∈ Z m is the natural projection.Similarly we define a R = (cid:112) π R ( π − E ( a )) (cid:113) ∈ R eq . Definition 3.3.
Given a definable set X in ( Z, R ) eq define X Z = { a Z : a ∈ X } where a (cid:55)→ a Z is given by Definition 3.2. Then X Z is a definable subset of somesort of Z eq and by stable embeddedness it is definable in Z eq . Similarly we define X R = { a R : a ∈ X } . Lemma 3.4.
Let X be a definable set in ( Z, R ) eq . Then X is R -internal if andonly if X Z is finite. Similary X is Z -internal if and only if X R is finite.Proof. Let X ⊆ S E = ( Z m × R n ) /E and suppose that X Z is finite. Unraveling thedefinitions this means that { π Z ( π E ( x )) − : x ∈ X } is a finite family of non-emptysubsets of Z m . So there is a finite subset b of Z m which meets all these sets. Thus X ⊆ π E ( b × R n ) ⊆ dcl( bR ) and X is R -internal.Conversely suppose that X is R -internal. Then { π Z ( π E ( x )) − : x ∈ X } isa family of subsets of Z m indexed by an R -internal set, so it must be finite byLemma 2.2. (cid:3) Lemma 3.5.
Let a ∈ ( Z, R ) eq . Then a is algebraic over ( a Z , a R ) ∈ Z eq × R eq .Proof. By Lemma 3.4 the set { x | x Z = a Z } ∩ { x | x R = a R } is internal to both Z and R , and therefore it is finite by Lemma 2.4 . Since a belongs to this finite set,the Lemma is established. (cid:3) Definition 3.6.
Given a ∈ ( Z, R ) eq define a Z = acl( a ) ∩ Z eq . More generally, givena set A of parameters from ( Z, R ) eq , define A Z = acl( A ) ∩ Z eq . Similarly define a R and A R . Observation 3.7.
Let a ∈ ( Z, R ) eq . Then:(1) acl( a Z ) ∩ Z eq = acl( a ) ∩ Z eq = a Z .(2) a is definable over a Z ∪ a R .Proof. By Lemma 3.5 we have acl( a ) = acl( a Z a R ), and by Lemma 2.5(4) the ele-ment a R ∈ R eq does not contribute to the algebraic closure in Z eq , so the first partis established. The second part is Lemma 2.5(2) in the new notation. (cid:3) A. BERARDUCCI AND M. MAMINO dimension of types In the sequel we assume that the theory of Z is superstable of finite SU-rankand that the SU-rank is definable. In Z we have a good notion of dimension givenby the SU-rank, and we want to define a dimension on ( Z, R ) eq . We recall thefollowing: Fact 4.1.
Let M be a superstable structure of finite SU -rank. Then:(1) (Additivity) For every a, b ∈ M eq and every set A of parameters from M eq we have SU( ab/A ) = SU( a/A ) + SU( b/aA ) .(2) (Definability) If M has SU -rank , then the SU -rank is definable, namelyfor every definable family ( X t ) t ∈ Y of sets in M eq and every k ∈ N , the set { t ∈ Y : dim( X t ) = k } is definable. Part (1) is well known and follows from Lascar’s inequalities [9, Theorem 8]. Forpart (2) see [13, Corollary 5.11].
Definition 4.2.
Given a ∈ ( Z, R ) eq definedim( a/A ) = SU( a Z /A Z ) , where A Z = acl( A ) ∩ Z eq (Definition 3.6) and a Z is as in Definition 3.2. Lemma 4.3.
Work in ( Z, R ) eq . We have:(1) If b ∈ acl( A ) , then dim( b/A ) = 0 .(2) dim( ab/A ) = dim( a/A ) + dim( b/aA ) .(3) If r ⊂ R eq , then dim( a/Ar ) = dim( a/A ) .Proof. (1) Assume b ∈ acl( A ). Then b Z ∈ A Z and dim( b/A ) = SU( b Z /A Z ) = 0.Part (2) follows from the additivity formula for the SU-rank in Z eq , observingthat aA Z = acl( a Z A Z ), where the algebraic closure is inside Z eq .For (3) it suffices to observe that Ar Z = A Z . (cid:3) Proposition 4.4.
Let a be an element of ( Z, R ) eq and A ⊆ B be sets of parame-ters in ( Z, R ) eq . Then there is b ∈ ( Z, R ) eq such that dim( b/B ) = dim( a/A ) and tp( b/A ) = tp( a/A ) .Proof. By Lemma 3.5 there is an algebraic formula φ ( x, a Z , a R ) (possibly with ad-ditional parameters from A ) which isolates the type of a over a Z a R A . In particularfor each ψ ( x ) ∈ tp( a/A ) we have: | = ∀ x ( φ ( x, a Z , a R ) → ψ ( x )) . ( † )Let e ∈ Z eq be a realization of a non-forking extension of tp( a Z /A Z ) to B Z . Thismeans that tp( e/A Z ) = tp( a Z /A Z ) and SU( e/B Z ) = SU( e/A Z ). By Lemma 2.5(4)we have tp( e/A ∪ R ) = tp( a Z /A ∪ R ). It follows that we can replace a Z with e in( † ) and obtain | = ∀ x ( φ ( x, e, a R ) → ψ ( x )) . We have thus proved that φ ( x, e, a R ) isolates tp( a/A ). Moreover φ ( x, e, a R ) is analgebraic consistent formula, since this fact is a property in the type of e inheritedfrom a Z . Choose b ∈ ( Z, R ) eq such that φ ( b, e, a R ) holds. We can assume that φ ( x, u, v ) implies u = x Z and v = x R , since otherwise we can add these conditionsto the formula (using the fact that x (cid:55)→ x Z and x (cid:55)→ x R are definable functions).So b Z = e and b R = a R . To conclude it suffices to observe that dim( b/B ) =SU( b Z /B Z ) = SU( e/B Z ) = SU( e/A Z ) = SU( a Z /A Z ), where the last equalityfollows from the fact that tp( e/A Z ) = tp( a Z /A Z ). (cid:3) Note that in the above Lemma the sets of parameters A and B must be small(with respect to the monster model C ), so we cannot take B = Z say. We cannevertheless obtain the following corollary, where Z ≺ Z is a small model, but R = R ( C ) is interpreted in the monster model. Corollary 4.5.
Let a be an element of ( Z, R ) eq . Then there is b ∈ ( Z, R ) eq suchthat dim( b/a ) = dim( a ) and tp( b/ acl( Z ∪ R )) = tp( a/ acl( Z ∪ R )) . The idea is to use the fact that the type of a over the big set of parameters R isimplied by the type of a over the small set a R ⊂ R eq . The details are as follows. Proof.
By Observation 3.7 we have a R = acl( a ) ∩ R eq = acl( a R ) ∩ R eq . By Lemma4.4 (with A = a R and B = a R ∪ a ) there is some b with tp( b/a R ) = tp( a/a R )and dim( b/a R ∪ a ) = dim( a/a R ). By Lemma 4.3 the parameters from R eq do notcontribute to the dimension, namely dim( a/a R ) = dim( a ) and dim( b/a R ∪ a ) =dim( b/a ). It remains to show that tp( b/ acl( Z ∪ R )) = tp( a/ acl( Z ∪ R )). Wecan assume that the parameters from Z are named by constants in the language,so we only need to prove worry about R . So assume φ ( x, r ) ∈ tp( a/R ), where r ∈ R eq , and let us prove that φ ( x, r ) ∈ tp( b/R ). By Lemma 2.5(6) there is aformula ψ ( x ) ∈ tp( a/a R ) which implies φ ( x, r ). Since tp( b/a R ) = tp( a/a R ) wededuce that φ ( b, r ) holds. (cid:3) Dimension of definable sets
Definition 5.1.
Given a definable set X in ( Z, R ) eq letdim( X ) = max a ∈ X dim( a/A )where A is any set of parameters over which X is defined and a ranges in themonster model. By Proposition 4.4 this does not depend on the choice of A . Proposition 5.2.
Let M be a model of a superstable theory of finite SU -rank. Let X be a definable subset of M n and let Y be an M -definable subset of X of the same SU -rank. Suppose that X is defined using parameters in a model M ≺ M . Then Y has a point with coordinates in acl( M ) . The same holds for definable sets in M eq provided the SU -rank is definable.Proof. Let m = SU ( X ) = SU ( Y ). By definition SU( Y ) = sup { SU ( b/M ) : b ∈ Y } = m . By our assumption m is a finite ordinal, so the sup is achieved. Choose b ∈ Y with SU( b/M ) = m . Then SU( b/M ) ≥ SU ( b/M ) = m . On the otherhand m = SU ( X ) ≥ SU ( b/M ). Thus tp ( b/M ) has the same SU-rank of tp ( b/M ).If X is included in one of the real sorts M n , this implies that tp ( b/M ) is finitelysatisfiable in M . (Indeed for types over models in a superstable theory, the uniqueextension with the same SU-rank coincides with the unique non-forking extension,which in turn coincides with the unique unique extension finitely satisfiable in thesmall model). Thus in this case Y has a point with coordinates in M .The case when X is included in one of the imaginary sorts M n /E is easilydeduced from the real case assuming the definability of the SU-rank. To see this, let D i ⊆ M n be the union of all the E -equivalence classes of SU-rank i . By definabilityof the SU-rank, the set D i is definable and SU( X ) = max i SU( X ∩ π E ( D i )). So wecan reduce to the case when all the E -equivalence classes have the same SU-rank.Next observe that if all the equivalence classes have SU-rank i , then by additivityof the SU-rank we have dim( X ) = dim( π − E ( X )) − i and similarly for Y . By thereal case we deduce that π − E ( Y ) intersects acl( M ) eq , hence so does Y . (cid:3) Remark 5.3.
Reasoning as in the last part of the proof, the definability of the SU-rank for definable families in the imaginary sorts of M eq follows (using additivity)from the definability of the SU-rank for definable families in the home sorts M n . Theorem 5.4.
Assume that the theory of Z is superstable of finite SU -rank thatthat the SU -rank is definable. Put in ( Z, R ) eq the dimension function of Definition5.1. We have:(1) (Additivity) Given a definable surjective function f : X → Y with fibers ofconstant dimension k , we have dim( X ) = dim( Y ) + k .(2) (Monotonicity) dim( X ∪ Y ) = max(dim( X ) , dim( Y )) .(3) (Base) dim( X ) = SU ( X Z ) .(4) (Dimension zero) dim( X ) = 0 if and only if X is R -internal.(5) (Density) If X is ∅ -definable and Y ⊆ X is a definable subset of the samedimension, then for every Z ≺ Z the intersection of acl( Z ∪ R ) eq with Y is non-empty.(6) (Definability) Given d ∈ N and a definable family ( X t : t ∈ Y ) definedover A , the set Y d := { t ∈ Y | dim( X t ) = d } is definable over A .Proof. (1) Assume for simplicity that f : X → Y is ∅ -definable in ( Z, R ) eq andconsider an element b ∈ Y with dim( b ) = dim( Y ) and an element a ∈ f − ( b ) withdim( a/b ) = dim( f − ( b )) = k . Thendim( X ) ≥ dim( a ) = dim( ab ) (by Lemma 4.3)= dim( b ) + dim( a/b )= dim( Y ) + k. Similarly, starting an element a ∈ X with dim( a ) = X and letting b = f ( a ) weobtain the opposite inequality.(2) is obvious.(3) This follows from the additivity property applied to the definable function x (cid:55)→ x Z after observing that for each a ∈ X Z the fiber { x ∈ X : x Z = a } is R -internal by Lemma 3.4, and therefore it has dimension zero by the easy part of(4).(4) Assume dim( X ) = 0. Then SU( X Z ) = 0. So X Z is finite and by Lemma 3.4 X is R -internal. The converse is clear.(5) The corresponding property for the SU-rank in Z eq is given by Lemma 5.2.Thanks to (3) we can reduce to the Z eq -case replacing X with X Z and Y with Y Z . Indeed we only need to observe that if Y Z meets acl( Z ∪ R ) then so does Y (because y ∈ acl( y Z , y R ) by Lemma 3.5).(6) Given a definable family ( X t : t ∈ Y ) in ( Z, R ) eq we have dim( X t ) =SU( X tZ ), so we need to prove that SU( X tZ ) = d is a definable condition on t .Since X tZ is definable in Z eq this is almost given by our assumptions on Z . Theonly problem is that the parameter t does not range in Z but in a sort of ( Z, R ) eq .However, by stable embeddedness, the set ( X t ) Z can be defined by a formula φ ( x, b )with parameters b from Z . Moreover by a compactness argument there is a singleformula φ ( x, y ) and a definable function t (cid:55)→ b t ∈ Z m (for some m ∈ N ) such thatfor each t ∈ Y the set X tZ is defined by φ ( x, b t ). This reduces the definability ofdim in ( Z, R ) eq to the definability of SU in Z eq . (cid:3) The quantifier (Most x ∈ X ) Definition 6.1.
Given a definable set X in ( Z, R ) eq and a formula φ ( x ) we define: • (Few x ∈ X ) φ ( x ) ⇐⇒ dim( { x ∈ X : φ ( x ) } ) < dim( X ) • (Most x ∈ X ) φ ( x ) ⇐⇒ (Few x ∈ X ) ¬ φ ( x )From Theorem 5.4 we immediately deduce the following: Lemma 6.2.
Fix Z ≺ Z . We have:(1) | = (Most x ) φ ( x ) ∧ (Most x ) ψ ( x ) ⇐⇒ (Most x )( φ ( x ) ∧ ψ ( x )) . (2) If f : X → X is a definable bijection, then | = (Most x ∈ X ) φ ( x ) ⇐⇒ (Most x ∈ X ) φ ( f ( x )) .(3) Given a formula φ ( x, y ) there is a formula ψ ( y ) such that | = (Most x ∈ X ) φ ( x, y ) ⇐⇒ ψ ( y ) .(4) Let φ ( x ) be a formula, possibly with parameters, and let X be ∅ -definable in ( Z, R ) eq . Suppose that for all points a ∈ X algebraic over Z ∪ R we have φ ( a ) . Then | = (Most x ∈ X ) φ ( x ) .Proof. We apply Theorem 5.4. Part (1) follows from the additivity of dimension.Part (2) from the monotonicity. Part (3) from the definability. Point (4) from thedensity property. (cid:3) Main theorem
We are now ready to prove our main result.
Theorem 7.1.
Let Z be a superstable structure of finite SU -rank and assume thatthe SU -rank is definable. Let R be an arbitrary structure. Given a group ( G, · ) definable in ( Z, R ) eq , there is a Z -internal definable normal subgroup Γ (cid:67) G suchthat G/ Γ is R -internal.Proof. Let S E = ( Z m × R n ) /E be the sort of G . Given x ∈ G , let R ( x ) = π R ( π − E ( x )) ⊆ R m where π R : Z m × R n → R n is the projection. Note that x R = (cid:112) R ( x ) (cid:113) according to Definition 3.2. DefineΓ = (cid:8) g ∈ G : (Most y )(Most x ) (cid:0) R ( xg y ) = R ( g y x ) = R ( x ) (cid:1)(cid:9) where g y = ygy − and (Most − ) stands for (Most − ∈ G ).To prove that Γ is a subgroup we use the invariance of the quantifier (Most − ) un-der bijections and the fact that the quantifier distributes over conjunctions (Lemma6.2). The details are as follows. Let a, b ∈ Γ. We need to show that ab ∈ Γ. Wehave: b ∈ Γ= ⇒ (Most y )(Most x ) (cid:0) R ( xb y ) = R ( x ) (cid:1) = ⇒ (Most y )(Most x ) (cid:0) R ( xa y b y ) = R ( xa y ) (cid:1) since x (cid:55)→ xa y is a bijection= ⇒ (Most y )(Most x ) (cid:0) R ( x ( ab ) y ) = R ( xa y ) (cid:1) On the other hand a ∈ Γ= ⇒ (Most y )(Most x ) (cid:0) R ( xa y ) = R ( x ) (cid:1) . So combining the two derivations we obtain a, b ∈ Γ= ⇒ (Most y )(Most x ) (cid:0) R ( x ( ab ) y ) = R ( x ) (cid:1) and similarly one obtains a, b ∈ Γ= ⇒ (Most y )(Most x ) (cid:0) R ( x ( ab ) y ) = R ( x ) = R (( ab ) y x ) (cid:1) witnessing ab ∈ Γ.Let us now prove that a ∈ Γ implies a − ∈ Γ. (Here is where we need the factthat in the definition of Γ we have both xg y and g y x .) We have: a ∈ Γ= ⇒ (Most y )(Most x ) (cid:0) R ( xa y ) = R ( x ) (cid:1) = ⇒ (Most y )(Most x ) (cid:0) R ( x ) = R ( x ( a − ) y ) (cid:1) where in the last implication we used the fact that x (cid:55)→ x ( a − ) y is a bijection.Similarly we obtain | = (Most y )(Most x ) (cid:0) R ( x ) = R (( a − ) y x ) (cid:1) . Together with the previous condition this yields a − ∈ Γ.We have thus proved that Γ is a subgroup. Let us now check that Γ is normalin G . To this aim note that b ∈ Γ can be expressed in the form (Most y ) Q ( b y )where Q is a suitable formula. If z ∈ G we want to show that b z ∈ Γ, namely(Most y ) Q (( b z ) y ) holds. This follows from the fact that ( b z ) y = b ( zy ) and y (cid:55)→ zy is a definable bijection on G .We need to prove that Γ is Z -internal. Without loss of generality we can work ina ℵ -saturated model. Suppose for a contradiction that Γ is not Z -internal. Thenthere is a countable infinite subset { g i : i ∈ ω } of Γ such that R ( g i ) (cid:54) = R ( g j ) for i (cid:54) = j (Lemma 3.4). By ℵ -saturation and the definition of Γ, there are x, y ∈ G suchthat for every i ∈ ω we have R ( xg yi ) = R ( x ). Fix such an x, y and let f : G → G be the definable bijection g (cid:55)→ xg y . We then have R ( f ( g i )) = R ( x ) for all i ∈ ω ,namely each g i belongs to the definable set S := { g ∈ G : R ( f ( g )) = R ( x ) } . Thisset is in definable bijection with S (cid:48) := { g ∈ G : R ( g ) = R ( x ) } so it is Z -internal byLemma 3.4. On the other hand, still by Lemma 3.4, S cannot be Z -internal sinceit contains the infinite sequence { g i : i ∈ ω } and R ( g i ) (cid:54) = R ( g j ) for all i (cid:54) = j . Thiscontradiction shows that Γ is Z -internal.It remain to show that G/ Γ is R -internal, or equivalently (by Theorem 5.4) thatdim( G/ Γ) = dim( G ) − dim(Γ) = 0. We can assume that G is ∅ -definable. Let a ∈ G be such that dim( a ) = dim( G ). By Corollary 4.5 there is b ∈ G be suchthat tp ( b/ acl( Z ∪ R )) = tp ( a/ acl( Z ∪ R )) and dim( b/a ) = dim( a ). It followsthat dim( ab − ) = dim( G ) (because dim( ab − ) ≥ dim( ab − /a ) = dim( b − /a ) =dim( b/a ) = dim( a ) = dim( G )). We claim that ab − ∈ Γ. Granted this we havedim(Γ) ≥ dim( ab − ) and we obtain dim( G ) = dim(Γ) as desired. To prove theclaim we reason as follows. Since a, b ∈ G have the same type over acl( Z ∪ R ), forall x, y ∈ G ∩ acl( Z ∪ R ) we must have R ( a y x ) = R ( b y x ) . Indeed if R ( a y x ) (cid:54) = R ( b y x ), then taking r ∈ R in the symmetric difference of R ( a y x )and R ( b y x ), we obtain a formula with parameters in r, x, y which distinguishes thetypes of a and b .By Lemma 6.2 this implies | = (Most y )(Most x ) (cid:0) R ( a y x ) = R ( b y x ) (cid:1) Hence, by the definable bijection x (cid:55)→ ( b − ) y x we also get | = (Most x )(Most x ) (cid:0) R (( ab − ) y x ) = R ( x ) (cid:1) . By the same method we obtain | = (Most y )(Most x ) (cid:0) R ( x ( ab − ) y ) = R ( x ) (cid:1) . namely ab − ∈ Γ. (cid:3) Corollaries
In this section we study the subgroup Γ · C G (Γ) of G and we show that it coincideswith G if G is connected (i.e. it has no definable subgroups of finite index). Weneed the following. Lemma 8.1.
Let f : X → Y be a definable surjective function in ( Z, R ) eq andsuppose that Y is R -internal. Then there is an R -internal definable subset U of X such that f | U : U → Y is surjective. Proof.
We can assume that X ⊆ Z m × R n (if X is a subset of some imaginary sort( Z m × R n ) /E we reduce to this case by considering f ◦ π E : Z m × R n → Y ). For t ∈ Z m , let X t denote X ∩ { t } × R n . Then X t is R -internal and by Remark 2.2 theset { f ( X t ) : t ∈ Z m } is finite. It follows that there is a finite subset Γ of Z m suchthat (cid:83) t ∈ Γ f ( X t ) = (cid:83) t ∈ Z m f ( X t ) = Y . So we can take U = (cid:83) t ∈ Γ X t . (cid:3) Lemma 8.2.
Let → A → B f → C → be an exact sequence of definable grouphomomorphisms in ( Z, R ) eq and assume that A and C are R -internal. Then B is R -internal.Proof. By Lemma 8.1 there is an R -internal subset U of B such that f | U : U → C is surjective. Thus B = U · ker( f ). Since U and ker( f ) are R -internal, it followsthat B is R -internal. (cid:3) We can now obtain the following corollary from the main theorem.
Corollary 8.3.
Under the hypothesis of Theorem 7.1 we have:(1)
G/C G (Γ) is Z -internal.(2) Γ · C G (Γ) is a subgroup of finite index of G .(3) If G has no non-trivial Z -internal quotients (this may be regarded as anotion of connectedness), then Γ is included in the center of G .(4) G/Z (Γ) is a direct product of an R -internal and a Z -internal subgroup.Proof. (1) Consider the action of G on Γ by conjugation. This gives a morphismfrom G to Aut (Γ) whose kernel is C G (Γ). Thus G/C G (Γ) can be identified with adefinable family of automorphisms of Γ, and since Γ is Z -internal it easily followsthat G/C G (Γ) is Z -internal.(2) The group G Γ · C G (Γ) is both Z -internal and R -internal (being a quotient ofboth G/ Γ and
G/C G (Γ)), so it must be finite.(3) If G has no non-trivial Z -internal quotients, then G/C G (Γ) must be trivial(by (1)), so Γ is in the center of G .(4) We have Z (Γ) = Γ ∩ C G (Γ). So we have an isomomorphism G/Z (Γ) ∼ = G/ Γ × G/C G (Γ) and point (4) is established. (cid:3) O-minimal case
In this section we show (Theorem 9.7) that if R is an o-minimal structure and Z is arbitrary, then every group G definable in ( Z, R ) admits a unique “ t -topology”in analogy with the o-minimal case [12]. In particular, if R is based on the reals,then G has a natural structure of a real Lie group. Moreover any Z -internal subsetof G is discrete in the t-topology. If we additionally assume that Z is superstableof finite Lascar rank we can then apply Theorem 7.1 to show that any group G definable in ( Z, R ) is a cover of a group definable in R (Corollary 9.14). Here by“cover” we mean a definable morphism which is also a local homeomorphism in thet-topologies (we do not require G to be connected).In § Z, R ) eq based on the projection on the Z -coordinates and using the SU-rank on Z . In this section we introduce anotherdimension based on the o-minimal dimension on R . Throughout the section, withthe exception of Corollary 9.14, R is o-minimal and Z is arbitrary. Definition 9.1.
Given a definable set X ⊆ Z m × R n , we define dim R ( X ) as theo-minimal dimension of the projection of X to R n . For X definable in ( Z, R ) eq wedefine dim R ( X ) as the o-minimal dimension of X R , where X R is as in Definition 3.3. Proposition 9.2.
Work in ( Z, R ) eq , where R is o-minimal. We have: (1) (Additivity) Given a definable surjective function f : X → Y with fibers ofconstant R -dimension k , we have dim R ( X ) = dim R ( Y ) + k .(2) (Monotonicity) dim R ( X ∪ Y ) = max(dim R ( X ) , dim R ( Y )) .(3) (Dimension zero) dim R ( X ) = 0 if and only if X is Z -internal.(4) (Definability) Given d ∈ N and a definable family ( X t : t ∈ Y ) definedover A , the set Y d := { t ∈ Y | dim R ( X t ) = d } is definable over A .Proof. This is similar, mutatis mutandis , to the proof of the corresponding points inTheorem 5.4, but with the roles of Z and R exchanged. Indeed the relevant pointsof Theorem 5.4 (namely everything with the exception of the “Density” property)do not use the stability assumption, but only the fact that we have a good notionof dimension on the relevant sort. So we can carry out exactly the same argumentswith the o-minimal dimension instead of the SU -rank. (cid:3) Definition 9.3.
Given a definable set X and a definable subset Y ⊆ X , we saythat Y is R -large in X if dim R ( X \ Y ) < dim R ( X ) (with the convention that ifdim R ( X ) = 0 this means that X = Y ). Equivalently, Y is R -large in X if Y intersects every R -internal subset of X of maximal dimension into a large set. Wesay that a point a ∈ X is R -generic over a set of parameters A if a belongs to every R -large subset of X defined over A . Lemma 9.4.
Let G be a definable set in ( Z, R ) with R o-minimal. If X ⊆ G is R -large in G , then X is generic, namely finitely many translates of X cover G .Proof. Choose a small model M over which G is defined. It suffices to provethat G = G ( M ) · X . By orthogonality G has an R -internal definable subset Y of maximal R -dimension which is defined over M (one of the sections over the Z -coordinates). Now if U is R -large in G , then U ∩ Y is large in Y , and since Y is R -internal, it contains a tuple from M . We have thus proved that every R -large subset of G contains a point from M . To finish the proof, let X ⊆ G be R -large and let us show that G = G ( M ) · X . To this aim let g ∈ G . Then g · X − is R -large in G by the invariance of dim R under definable bijections (whichfollows from Proposition 9.2), hence it has a point γ from M . It follows that g ∈ γ · X ⊆ G ( M ) · X and since g ∈ G was arbitrary we obtain G = G ( M ) · X . (cid:3) Definition 9.5.
Let R be o-minimal. We put on R the topology generated by theopen intervals, and on Z the discrete topology. Given a subset X ⊆ Z m × R n , wedefine the ambient topology on X as the topology it inherits from Z m × R n , wherethe cartesian powers have the product topology.When X is a definable group, in addition to the ambient topology, we willintroduce also a group topology on X , called the t-topology , which will be definedby a suitable modification of the construction in [12]. See also [10, 16, 7] for similarresults. Our proof is self contained. Lemma 9.6.
Let X and Y be definable sets in the two-sorted structure ( Z, R ) . Let f : X → Y be a definable function. Then the set of continuity points of f , withrespect to the ambient topology, is R -large in X .Proof. The result is well known in the o-minimal case and since the notion of R -largeness refers to the R -internal sets we can readily reduce to that case. (cid:3) Theorem 9.7.
Let G be a definable group in ( Z, R ) , with R o-minimal and Z arbitrary. Then G has a unique group topology, called the t-topology, which coincideswith the ambient topology on an R -large open subset V of G .Proof. By Lemma 9.6 there is an R -large subset Y of G × G × G such that thefunction α : ( x, y, z ) (cid:55)→ xyz from G × G × G to G is continuous on Y (with respect to the ambient topology). Replacing Y with its interior we can assume that Y is open in G × G × G (still with respect to the ambient topology). Note that if( x, y, z ) ∈ G × G × G is generic, then it belongs to Y . Let V be the set of all x ∈ G such that for all ( g , g ) ∈ G × G which are R -generic over x both ( g , x, g )and ( g , g − xg − , g ) belong to Y . By the definability of dimension V is definable.Moreover V contains all R -generic elements, so it is R -large in G . Let V be theinterior of V in G . Then V is definable, R -large, and open in G . Claim 1.
For all a, b ∈ G , Z := V ∩ aV b is open in V , and the function f : x (cid:55)→ a − xb − from Z to V is continuous. Proof . Let z ∈ Z . We will show that there is an open neighbourhood of z con-tained in Z and f is continuous at z . Pick ( a , b ) ∈ G × G generic over a, b, z .Write a − = a a and b − = b b . We write f as the composition of x g (cid:55)→ a xb h (cid:55)→ a − xb − where h : ζ (cid:55)→ a ζb . Consider the following subsets of ZZ := { ζ ∈ V | ( a , ζ, b ) ∈ Y } Z := { ζ ∈ Z | ( a , a ζb , b ) ∈ Y } Clearly Z is open in V , g is continuous on Z , and z ∈ Z . From this it followsthat Z is open in V and f is continuous on Z . Moreover z ∈ Z observing that( a , a zb , b ) = ( a , a − a − zb − b − , b ) and ( a , b ) is generic over a, b, z . Now, f | − Z ( V ) ⊆ Z is an open neighbourhood of z , and f is continuous at z . (cid:3) claim We define the t-topology. A subset O of G is t-open iff for all a, b ∈ G the sub-set aOb ∩ V of V is open in V .By the previous claim V is t-open. More generally we have: Claim 2. O ⊆ V is open in G (with respect to the ambient topology) if and only ifit is t-open. In other words the t-topology and the ambient topology coincide on V . Proof . Clearly a t-open subset of V is open in V . Conversely suppose that O ⊆ V is open in V . We must prove that aOb ∩ V is open in V . We have aOb ∩ V = a ( O ∩ a − V b − ) b = aO (cid:48) b where O (cid:48) := O ∩ a − V b − . Note that O (cid:48) is an open subsetof V because it can be written as the intersection of V ∩ a − V b − (which is openby the previous claim) and O . To prove that aO (cid:48) b is open it suffices to observe that aO (cid:48) b = f − ( O (cid:48) ) where f is the continuous function of the previous claim. (cid:3) claim Now we prove that the group operation is t-continuous. The group translationsare clearly t-continuous. Since V is R -large, there are a, b, c ∈ V such that ab = c .By Lemma 9.6 the group operation µ is continuous at ( a, b ) with respect to theambient topology, hence also t-continuous (since the two topologies coincide on V ).To prove t-continuity at another point ( x, y ) ∈ G × G , we go from ( x, y ) to ( a, b )by the t-continuous map ( ax − ( · ) , ( · ) y − b ), then from ( a, b ) to ab by µ , and finallyfrom ab to xy via xa − ( · ) b − y .Finally we show that the group inverse x (cid:55)→ x − is t-continuous. Consider an R -generic point a ∈ G . Then a − is also R -generic, so both a and a − belong to V .By Lemma 9.6 the group inverse is continuous at a , hence t-continuous. To provecontinuity at another point b ∈ G , note that x (cid:55)→ ( xb − a ) − is t-continuous at b and x − = b − a ( xb − a ) − is obtained by composing with a group translation.We have thus proved the existence of a group topology which coincides withthe ambient topology on an R -large open subsets. Granted the existence, theuniqueness is clear. (cid:3) In analogy with the o-minimal case we have:
Proposition 9.8.
Let G be a definable group in ( Z, R ) with R o-minimal. Everydefinable subgroup H of G is closed in the t-topology. Proof.
First note that the t-closure of a definable subset of G is definable. Indeedthe closure of a definable set in the ambient space is definable and we can reduceto this case working in an R -large subset V of G where the two topologies coincide.So, replacing H with its closure, it suffices to show that a dense subgroup H of G coincides with G . So assume that H is dense. By o-minimality of R , a densesubset of V has interior. It follows that H has interior and therefore it is t-open in G . Being also a dense subgroup, it coincides with G . (cid:3) We next show that the t-topology of a quotient coincides with the quotienttopology.
Theorem 9.9.
Let f : G → H be a definable surjective homomorphism of definablegroups in ( Z, R ) , with R o-minimal. Then f is continuous and open with respect tothe t-topologies. So the t-topology of H can be identified with the quotient topologyof G/ ker( f ) . We need the following easy lemma, true in arbitrary o-minimal structures:
Lemma 9.10.
Work in an o-minimal structure. Let f : X → Y be a definablecontinuous surjective map. Then there is an open subset U of X such that f | U : U → Y is an open map.Proof. Let G ( f ) be the graph of f . Then X is definably homeomorphic to G ( f ).Replacing X with G ( f ) we can assume that f is a projection on some coordinates.So we must prove that if Z is a definable subset of X × Y and π : Z → X is theprojection on X , then there is an open subset U of Z such that f | U : U → X is anopen map. To this aim take a cell decomposition of Z such that the projections ofthe cells of Z give a cell decomposition of X . Let C be a cell of maximal dimensionof X . Then π − ( C ) is an open subset of Z and π restricted to this set is an openmap. (cid:3) Proof of Theorem 9.9.
Consider the restriction of f to an open R -internal subset U . By o-minimality f | U is continuous at some point. Since the group translationsare continuous, f is continuous everywhere. We prove that f is open. First notethat, by the additivity of dimensions, if Y is a definable subset of H then dim R ( Y )+dim R (ker( f )) = dim R ( f − ( Y )). Considering the complements it follows that theimage of an R -large subset of G is R -large in H . To prove that f is open it sufficesto prove that it is open at some point. Consider an open large subset U of G .Then f ( U ) is large in H and therefore it has interior in H (since H has an R -largeopen subset in the t-topology and two R -large sets have a non-empty intersection).We can then take a definable subset V of f ( U ) which is open in H and considerthe restriction f | f − ( V ) : f − ( V ) → V . By Lemma 9.10 f | f − ( V ) is open atsome point in the ambient topologies of its domain and image. But the t-topologycoincide with the ambient topology on this sets, so f is open at some point in thet-topology. (cid:3) We can now introduce a notion of connectedness for definable groups in (
Z, R ).This motivates the parenthetical remark in Corollary 8.3(3).
Definition 9.11.
Let G be a definable group in ( Z, R ). We say that G is t-connected if G satisfies one of the equivalent conditions of Proposition 9.12 below. Proposition 9.12.
Let G be a definable group in ( Z, R ) , with R o-minimal. Thefollowing are equivalent:(1) G has no proper clopen definable subsets in the t-topology.(2) G has no proper open definable subgroups in the t-topology.(3) G has no definable subgroups H such that G/H is Z -internal. (4) G has no definable subgroups H with dim R ( H ) = dim R ( G ) .Proof. Let us first prove the equivalence of (1) and (2). So let X is a clopen definablesubset of G . It suffices to show that its stabilizer Stab( X ) = { g : gX ⊆ X } is anopen subgroup of G (note that Stab( X ) (cid:54) = G if ∅ (cid:54) = X (cid:54) = G ). To this aim it sufficesto observe that any t-connected open neighborhood of the identity of G must becontained in Stab( X ).To finish the proof it suffices to prove the equivalence of the following conditions:(i) H is an open subgroup in the t-topology of G .(ii) dim R ( H ) = dim R ( G ).(iii) G/H is Z -internal.To show that (i) implies (ii), let V be a large subset of G where the ambienttopology coincides with the t-topology. If H < G is open, then H intersects V is an open set, so it has maximal R -dimension. The implication from (ii) to (i) iseasy. The equivalence of (ii) and (iii) follows from the additivity of the R -dimension(Proposition 9.2). (cid:3) Remark 9.13.
Every Z -internal subset of G is discrete in the t-topology. Proof.
The t-topology has a basis of R -internal sets. Since an R -internal set canintersect a Z -internal set in at most finitely many points, each Z -internal subset isdiscrete. (cid:3) So far in this section Z was an arbitrary structure. For the next corollary wealso need a stability assumption to be able to apply Theorem 7.1. Corollary 9.14. If R is o-minimal and Z is superstable of finite Lascar rank, thenevery group G definable in ( Z, R ) is a “cover” of a group definable in R , namelythere is a definable morphism f : G → H such that H definable in R and f is alocal homeomorphism (equivalently ker( f ) is Z -internal).Proof. By Theorem 7.1 there is a Z -internal subgroup Γ such that G/ Γ is inter-pretable in R (hence, by [14], definably isomorphic to a group H definable in R ).Moreover the morphism G → G/ Γ has a discrete kernel (by Remark 9.13) and it iscontinuous and open (by Theorem 9.9), so it is a local homeomorphism. (cid:3)
In the “classical” situation when R is an o-minimal expansion of the reals and Z = ( Z , +) it follows that every (connected) group G definable in (( Z , +) , R ) is acover, in the classical sense, of a real Lie group definable in R . The proof of thisresult was the intial motivation for our work.Let us finish by mentioning some related work on locally definable groups ino-minimal structures. In [6] it is proved that if G is a locally definable connectedabelian group in an o-minimal structure R and G exists, then G is a cover of adefinable group (and therefore it is interpretable in (( Z , +) , R ) by [8]). However inthe non-abelian case the corresponding result fails [1, Example 7.1], while Theorem9.14 holds also in the non-abelian case. In general the class of the locally definableconnected groups (even assuming that G exists) is much larger than the class ofgroups interpretable in (( Z , +) , R ). One way to see this is to observe that everygroup interpretable in (( Z , +) , R ) has the non-independence property (NIP), whileby [4] there are (connected) groups definable in (( Z , +) , ( R , + , · )) which interpretthe ring of integers (or even the real field with a predicate for the integers). References [1] Alessandro Berarducci, M´ario J. Edmundo, and Marcello Mamino. Discrete subgroupsof locally definable groups.
Selecta Mathematica [2] Alessandro Berarducci and Marcello Mamino. On the homotopy type of definable groups inan o-minimal structure.
Journal of the London Mathematical Society , 83(3):563–586, 2011.[3] Alessandro Berarducci, Ya’acov Peterzil, and Anand Pillay. Group covers, o-minimality, andcategoricity.
Confluentes Mathematici , 02(04):473–496, 2010.[4] Annalisa Conversano and Marcello Mamino. Private communication.[5] M´ario J. Edmundo and Pantelis E. Eleftheriou. The universal covering homomorphism ino-minimal expansions of groups.
Math. Log. Quart. , 53(6):571–582, November 2007.[6] Pantelis E. Eleftheriou and Ya’acov Peterzil. Definable quotients of locally definable groups.
Selecta Mathematica , 18(4):885–903, March 2012.[7] Antongiulio Fornasiero. Groups and rings definable in d-minimal structures. arXiv preprintarXiv:1205.4177 , pages 1–19, 2012.[8] Ehud Hrushovski, Ya’acov Peterzil, and Anand Pillay. On central extensions and definablycompact groups in o-minimal structures.
Journal of Algebra , 327(1):71–106, 2011.[9] Daniel Lascar. Ranks and definability in superstable theories.
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Oxford Logic Guides, 32. Oxford Science Publica-tions. The Clarendon Press, Oxford University Press, New York, 1996.[14] Janak Ramakrishnan, Ya’acov Peterzil, and Pantelis Eleftheriou. Interpretable groups aredefinable. arXiv:1110.6581 , pages 1–41, October 2011.[15] Katrin Tent and Martin Ziegler.
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