and
G
00
Mariana Vicar´ıa
Advisor: Alf Onshuus
1 Preliminaries 9
1.1 Some basic notions in Model theory . . . 11
1.1.1 Types and saturated models . . . 11
1.1.2 Dependent theories and indiscernible sequences . . . 14
1.2 Groups and measures in dependent theories . . . 18
1.2.1 Measures and Keisler Measures . . . 19
1.2.2 Keisler measures in dependent theories . . . 24
1.2.3 Groups with fsg . . . 25
1.3 Amenable Groups . . . 33
2 Analysis of definable groups in Presburger Arithmetic 45 2.1 Presburger Language and quantifier elimination . . . 46
2.1.1 Some consequences of quantifier elimination . . . 47
2.1.2 A-definable functions and dcl(A) . . . 47
2.2 Cell Descomposition theorem . . . 54
2.3 Dimension . . . 61
2.3.1 Definable Dimension . . . 61
2.3.2 Cell Dimension . . . 64
2.3.3 Open cells and boxes . . . 70
3 A characterization of definable groups in Presburger Arithmetic 77
3.1 Group operation vs usual addition . . . 77 3.2 Every definable group in Presburger is abelian-by-finite . . . 84 3.2.1 The centralizer connected component and the ICC property . . . . 84 3.2.2 G is abelian-by-finite . . . 87
4 Generic sets of definable groups in Presburger 91
4.1 Definable functions between two non- algebraic types . . . 92 4.2 The theorem of the finite intersection property and a point . . . 95 4.3 Generic sets of definable bounded groups . . . 107
5 Ellis semigroup conjecture 115
5.1 Topological dynamics . . . 115 5.2 The Ellis Group . . . 117 5.3 Topological dynamics from a model theoretic approach . . . 125
Introduction
In [7] (2008) Hrushovski, Peterzil and Pillay proved the following theorem:
Theorem 0.0.1. If G is a definably compact group definable in a saturated o-minimal
expansion of a real closed field, then the quotientG/G00 of Gby its smallest type-definable
subgroup of bounded index G00, is, when equipped with the logic topology, a compact Lie
group whose dimension (as a Lie group) equals the dimension of G (as a definable set in
an o-minimal structure).
In that paper these authors pointed out the connection between several topics, in or-der to classify the quotient group G/G00. The knowledge areas involved in their paper are amenability in dependent theories, the finitely satisfiable generics property (fsg), the generic sets and the existence of compact group H which dominates G, among others. Therefore, a natural question that might arise is: Can we generalize some of their results to other similar theories? How much all these theorems depend heavily on being in the o-minimal context?
In particular, we are interested in classifying the quotient group G/G00, where Gis a de-finable group in the theoryT =T h(Z,+, <), which is known as the Presburger Arithmetic theory. Actually, this theory has some properties which are very close to the o-minimal case. Indeed, it is a dependent theory and moreover it is quasi o-minimal. Additionally, R. Cluckers in [3] proved that there is a cell decomposition theorem for this theory, which allows us to define also a notion of dimension for the definable sets, such as in the o-minimal context.
However, we lose the topology induced on G by the order ≤, since in the Presburger case it corresponds to the discrete topology. This is a huge loss, indeed the topolog-ical tools, such as continuity and the connected components, were fundamental in the o-minimal context to show that G/G00 is a Lie group (see [14]).
Nevertheless, in this thesis we show that we can recover some of the results obtained in the o-minimal theories for the Presburger case. For example, following some ideas of P. Eleftheriou and S. Starchenko in [13] and using the cell decomposition theorem we can show that every definable group in Presburger is abelian-by-finite, which implies that it is also definably amenable.
Additionally, Peterzil and Pillay have shown that the definable non-generic subsets of a definably compact abelian group G, in an o-minimal theory, are an ideal. The crucial idea of this proof is presented in Theorem 2.1 of [8], and there is a similar statement of this theorem that holds for the Presburger case, actually we realized that the condition of being definably compact might be changed by being bounded. Moreover, this will be enough to show that any definable bounded group in Presburger has the fsg property, which implies that the Ellis group conjecture of Newelski hold for the definable bounded groups in Presburger.
In the first chapter we introduce all the necessary concepts, tools and theorems that are going to be relevant for this work. More precisely, we present a detailed explanation of amenable groups, a summary of dependent theories and their relationship with Keisler measures, an exposition of the fsg property and how it implies a characterization of G00 as a stabilizer. In the second chapter, we introduce the cell decomposition theorem devel-oped by R. Cluckers and we present two notions of dimension for the definable sets in the Presburger theory, one comes from a definable independence, while the second one arises from the cell decomposition theorem. Actually, we show that this two notions coincide. In the third chapter, we show that every definable group in Presburger is abelian-by-finite, while in Chapter 4 we present a similar statement of the Theorem 2.1 of [8], in order to show that every definable bounded group in Presburger has the fsg property. In Chapter
5, we present a proof developed by A. Pillay in [9] which establishes that the Ellis group conjecture of Newelski holds for every definable group with the fsg property.
Acknowledgements
Entregar esta tesis implica que me despido de los Andes, por lo que mis agradecimientos no los dirigir´e ´unicamente a cosas relativas a este trabajo. Esta tesis es simplemente el ´
ultimo paso, de muchos. Sin embargo, empezar´e por agradecerle a Alex y a Juan Felipe por sus ´utiles correcciones y detallada lectura.
Le debo a los Andes seis a˜nos de mucha mucha felicidad, y s´ı, claro que tengo nostal-gia. Me despido de las tardes en el s´otano haciendo las interminables tareas de Monika o de Ramiro, me despido del caf´e del departamento y de los cigarrillos compartidos al frente del H. Me despido de las guaridas secretas y los abrazos sigilosos. Me despido del f´utbol, de la copa Klein, de los goles tambaleantes y flojos. Le digo adi´os a los mi´ercoles de cerveza en el BBC y al bullying del grupo de l´ogica. Me despido de mi segunda casa. Me despido de profesores que admiro con toda la integridad del caso, por su dedicaci´on en sus clases y por hacer de mi aprendizaje algo siempre mejor. Gracias a Carlos, Monika y a Xavier por dejarme ver el gran cari˜no que sienten por lo que hacen y, en particular, por dejar algo de esa dedicaci´on en mi.
Me despido de mi asesor, y sobretodo mi amigo, Alf, a quien le estar´e eternamente agradecida por su incondicional apoyo y su confianza. Gracias por los caf´es, cigarril-los, goles, charlas, genialidades y falsedades de las que nos convencimos. Gracias por haber cre´ıdo m´as en m´ı de lo que yo creo. Gracias por la paciencia, y por mostrarme que las matem´aticas, m´as que mi deber, son algo que disfruto.
Tambi´en me despido de mis amigos, de las canciones pussy de Mate´ın, de los chistes os-curos de Lucho, del idealismo fascinante de Andy y del persistente color negro de Monika.
Me despido del inolvidable Sena de Samy, de las cervezas de Rafa, de la ligereza de Santi y la franqueza de Juan Felipe. Me despido de las tardes de billar con Dani, del gordelio de Alemanchis y de la versi´on hippie de Calder´on.
Me voy con l´agrimas en los ojos motivadas por una inmensa alegr´ıa de haberlos conocido, y anhelando que la vida eventualmente vuelva a ponernos en Magola. A muchos es posible que no los vuelva a ver, as´ı que: buen viento y buena mar.
Por supuesto, debo mencionar tambi´en el profundo agradecimiento que siento con mi fa-milia por su constante apoyo. En particular, debo agradecerle a mis pap´as por cada uno de los esfuerzos que hicieron a lo largo de su vida para darme la oportunidad de haber es-tado en los Andes. Les agradezco tambi´en haber visto mucho m´as all´a de las expectativas que ten´ıan para m´ı, y haberse preocupado sobretodo por mi felicidad. He de decirles, con toda la seguridad que puedo atribuirme, que fui extremadamente feliz. Gracias a ustedes ´esto fue posible.
Preliminaries
Notational convention
The main purpose of this section is to introduce all the notation that will be used through-out this thesis.
• Given sets X, Y, we sometimes denote the set of all functions f :X →Y by YX.
• Given a complete theory T, M will denote a saturated model and M0 will be a small elementary submodel of M.
• Gwill always denote a definable group in a complete theory T. Particularly, during Chapters 2 and 3, G will denote a definable group in the theory of Presburger Arithmetic.
• The notation a will denote a tuple of elements of M, and given a tuple a ∈ Mn,
say a= (a1, . . . , an) we use (a)i to denote the i-th coordinate of a, which isai.
• LetX ⊆ Mℓ and let 0 ≤n < ℓ, we indicate by π
≤n(X) the projection of X on the
first n coordinates, that is,
π≤n(X) :={(x1, . . . , xn)| (x1, . . . , xn, xn+1, . . . , xl)∈X}
• In the structures that we are going to study throughout this thesis we consider a linear order < that belongs to the language of Presburger Arithmetic. Thus, we introduce some standard notation for the intervals as follows:
– we indicate by [a, b] the set defined as{x∈ M | a≤x≤b},
– we denote by (a, b) the set defined as{x∈ M | a < x < b}.
• If X and Y are subsets of M we say that X < Y if and only if for every pair of elements x∈X and y∈Y we have thatx < y
• If G is a group and H is a subgroup, the cardinal [G :H] is called the index on H
in Gand denotes the number of left cosets of H inG.
• A system of linear congruences will be a finite set of equations of the form x ≡N
c, where 0 ≤ c < N are integers. Throughout this thesis, we will be using the generalization of the chinese reminder theorem, which says that a consistent linear system of congruences
x≡N1 c1, . . . , x≡Nk ck,
can be reduced to a unique equation x≡N c.
• Another result that we are going to recall several times istheErdos- Rado Theorem, which says:
Let κ be a cardinal and r, n∈N. For every partition of [κ]n={A⊆ κ : |A|=n}
intor pieces; then there is a subsetλ ≤κsuch that [λ]n is completely contained in
1.1
Some basic notions in Model theory
1.1.1
Types and saturated models
The main purpose of the first part of this section is to introduce several definitions that we are going to use through this thesis.
Definition 1.1.1 (An n-type over A). Fix a set of parameters A. An n-type over A is a consistent set π(x1, . . . , xn) of L(A)-formulas in n variables, x1, . . . , xn. We say that an
n-type p(x1, . . . , xn) is complete if it is a maximal consistent set.
Given ann-tupleb = (b1, . . . , bn) from a modelMand a subset of parametersA⊆ M,
we denote by tp(b/A) the complete n-type over A given by
tp(b/A) := {φ(x1, . . . , xn;a) :a ∈A and M |=φ(b1, . . . , bn;a)}.
Definition 1.1.2 (global type). LetMbe a very saturated model of a theoryT. Aglobal type is simply a completen-type p(x1, . . . , xn) over M.
Definition 1.1.3 (saturated model). Let T be a theory and M a model of T. We say thatMisκ-saturated if for every set of parameters A, such that|A|< κ, and anyn-type π(x1, . . . , xn) over A, we can findb1, . . . , bn∈ M such that:
M |=π(b1, . . . , bn)
Ifκ=|M|and Mis κ-saturated, then we say that Mis saturated.
Definition 1.1.4 (small set of parameters). If Mis a κ-saturated model, we say thatA is asmall set of parameters if|A|< κ. An elementary submodelM0, is asmall submodel of Mif |M0|< κ.
Definition 1.1.5 (finitely satisfiable type over M0). Let M0 be a small submodel of
π(x1, . . . , xn) isfinitely satisfiable overM0if for every finite subsetπ0 ={φi(x1, . . . , xn, a) :
i ≤ k} ⊆ π(x1, . . . , xn), there is a tuple (b1, . . . , bn) ∈ M0 such that M0 |= π0, that is,
M0 |=φ1(b1, . . . , bn, a)∧ · · · ∧φk(b1, . . . , bn, a).
Example 1.1.6. • LetT be the theory of the dense linear orders and let
M0 = (Q, <). Consider the type π(x) = {x > n | n ∈ N}, then p(x) is finitely satisfiable overM0: Note that for everyn1 <· · ·< nkinNwe have thatnk+1∈ M0 satisfies nk+ 1> n1 ∧nk+ 1> n2∧ · · · ∧nk+ 1 > nk.
• Let T be the theory of the fields of characteristic 0 and let p(x) = tp(i/R). This type is not finitely satisfiable over R, since the formula x2 + 1 = 0 ∈ p(x), but we know that there is no solution for the equation x2 + 1 = 0 in the field of real numbers.
Definition 1.1.7 (Invariant type). LetM0 be a small model such thatM0 ≺ M, where
M is a saturated model over M0. Let p(x) be a complete type overM we say that it is
M0-invariant if for any formula φ(x, b) and element b∈ M, wheter or not φ(x, b)∈p(x) depends only on the type tp(b/M0). In other words:
if tp(b/M0) =tp(b′/M0), then φ(x, b)∈p(x) if and only ifφ(x, b′)∈p(x).
Remark 1.1.8. LetM0 be an elementary submodel of Mand assume that Mis satu-rated overM0, and let p(x) be a complete type over M. Ifp(x) is finitely satisfiable over
M0 then it is alsoM0-invariant.
Proof. Letp(x) be a complete type overM, which is finitely satisfiable overM0, and let φ(x, y) be a formula. Letb, b′ be two elements such thattp(b/M0) =tp(b′/M0). In order to find a contradiction, assume thatφ(x, b)∈p(x) but¬(φ(x, b′))∈p(x). By completeness φ(x, b)∧ ¬(φ(x, b′))∈ p(x), and since p(x) is finitely satisfiable there is an element m of
M0 such that M φ(m, b)∧ ¬(φ(m, b
′
)). This implies that φ(m, y) ∈ tp(b/M0) and
Definition 1.1.9. Let I = hai | i ∈ ωi be a sequence of tuples in M and B ⊂ M a
set of parameters. We say that I is a B-indiscernible sequence if for every L(B)-formula φ(x1,· · · , xn;b) andi1 <· · ·< in, j1 < j2 <· · ·jn∈ω, we have that:
M |=φ(ai1,· · ·, ain;b)↔ M |=φ(aj1,· · · , ajn;b)
Example 1.1.10. Let T be the theory of dense linear orders without endpoints. Let
M= (Q, <), then:
• let p∈Q, the sequence defined as ai =p for all i∈ω is indiscernible.
• let a0 < a1 < . . . an increasing sequence of elements of Q, by the quantifier elimi-nation ofT, I =< ai |i∈ω > is indiscernible.
Definition 1.1.11 (EM-type). Let I = hai | i ∈ αi be a sequence, its
Ehrenfeucht-Mostowski (EM) type over a set of parameters B is the set of the L(B)- formulas that hold for every element ai ∈I.
EM(I/B) ={φ(x, b) | for all i∈α Mφ(ai, b)}
Definition 1.1.12 (Dividing formula). Let φ(xma) be a formula and A be a set of parameters. We say that φ(x, a) divides over A if there is an A-indiscernible sequence < ai | i < ω >(where a0 =a) such that the set:
{φ(x, ai) |i < ω}, (1.1)
is inconsistent.
say that φ(x, a) forks over A if
φ(x, a)←→
n _
i=1
ψ(x, bi), (1.2)
where each formula ψ(x, bi) divides over A.
Lemma 1.1.14. Let I = hai | i ∈ ωi be a sequence, then we can find an indiscernible
sequence C =hci | i∈ωi such that C satisfies the EM-type of I.
Proof. For a detailed explanation we refer the reader to [15] Lemma 5.13.
Definition 1.1.15 (Bounded Relation). Let X be a definable set. Assume that ∼ is an equivalence relation defined over X. We say that ∼ has boundedly many classes if
|X/∼ |<|M|, whereMis at least a 2|T|-saturated model. In this case, we also say that
∼ has boundedly many classes.
1.1.2
Dependent theories and indiscernible sequences
The main purpose of this section is to introduce some relevant facts about dependent theories. For a more detailed exposition see [10].
Definition 1.1.16 (Independence property and dependent formulas). Let φ(x, y) be an
L- formula, wherex is a k-tuple.
1. We say that φ(x, y) has the independence property if for any n ∈ ω, there are elements a1, . . . , an ∈ Mk such that for each subset S ⊆ {1,· · · , n}, there is an
element bS ∈ Ml, such that
M |=φ(ai, bS) if and only if i∈S.
2. We say that φ(x, y) is an dependent formula if it does not have the independence property.
3. An L-theoryT is adependent theory if any formula φ(x;y) is a dependent formula.
Example 1.1.17. • LetT be theory of dense linear orders and consider the formula φ(x, y) = x≤y.Observe that φ(x, y) is a dependent formula:
Let M be a saturated model of T, and fix two elements different p, q ∈ M. Let A = {p, q} and without loss of generality assume that p < q. Let b ∈ M be any element, then we have the following cases:
– if b < pthen A∩φ(M, b) =∅,
– if p≤b < q then A∩φ(M, b) ={p},
– if q≤b then A∩φ(M, b) ={p, q}
Therefore, we can not find some b ∈ M such that A∩φ(M, b) = {q}, hence this formula does not have the independence property.
• Let T be the theory of Peano Arithmetic and consider the formula φ(x, y) = x|y, which means ′′x divides y′′.Then φ(x, y) has the independence property:
Let N ∈N and consider A ={p0, . . . , pN−1} the firstN-primes. Let S be a subset of {0, . . . , N −1}, and let bS =
Y
i∈S
pi, therefore:
M |=φ(pi, bs) if and only if i∈S.
Therefore, φ(x, y) is an independent formula.
Example 1.1.18. The following are examples of dependent theories:
• Any o-minimal theory is dependent. [see [10], Chapter 2]
• Any quasi o-minimal theory is dependent, in particular Presburger Arithmetic is a dependent theory. [see [2]]
Definition 1.1.19 (shattered set). Let M be a model of a theory T. Let φ(x, y) be a formula, then we say that this formula shatters a subset A of M|x| if for any subset
S ∈ P(A), there is an element bS such that:
Mφ(a, bS)↔a∈S.
Remark 1.1.20. Letφ(x, y) be a formula, then it has the independe property if and only if for any n < ω there is some set A, which is shattered by φ(x, y).
Remark 1.1.21. A formula φ(x;y) has the independence property if and only if the formula ψ(y;x) := φ(x;y) has the independence property. That is, for every n < ω there are also elementsb1, . . . , bn∈ M|y|such that for eachS ⊆ {1,2, . . . , n}there isaS ∈ M|x|
such that M |=φ(aS;bi) if and only ifi∈S.
Theorem 1.1.22. Let φ(x, y) be a formula. This formula has the independence property
if and only if there is an indiscernible sequence hai | i∈ωiand a tuple b∈ Ml such that:
M |=φ(ai, b) if and only if i is even.
Proof. (⇐) Assume that there is a sequence hai | i ∈ ωi and a tuple b ∈ Ml as above.
LetI ⊆ω, we will show that there is somebI such thatM |=φ(ai, bI) if and only ifi∈I.
Claim: We can find an increasing one-to-one map τ : ω → ω such that for all i ∈ ω τ(i) is even if and only if i∈I
We will build this function by induction. If 0 ∈I, then we define τ0(0) = 0, otherwise we define τ0(0) = 1.
Suppose that we have constructed an injective map τn : {1,· · ·, n} →ω, then Im(τn) is
bounded in ω. Ifn+ 1∈I, then let m the minimal element inω\Im(τn) which is even,
and define τn+1 = τn∪ {(n+ 1, m)}. If n+ 1 ∈/ I, let τn+1 = τn∪ {(n+ 1, m+ 1)}. By
construction τn+1 is injective and increasing.
Letτ =Sn∈ωτn, thenτ is injective, increasing andτ(i) is even if and only ifi∈I. Claim Now, since hai | i ∈ ωi is an indiscernible sequence and τ is increasing, the map
sending each ai to aτ(i) is a partial isomorphism, and we can extend this function to a global automorphism σ of M. Let bI ∈ Ml such that σ(bI) =b.
Thus, M |= φ(ai, bI) if and only if M |= φ(aτ(i), b) if and only if τ(i) is even if and
only ifi∈I. We conclude that φ(x, y) has the independence property.
(⇒) Assume that φ(x, y) has the independence property, and let A = hai | i ∈ ωi
be a sequence of |x|-tuples which is shattered by φ(x, y). By Ramsey and compactness (Lemma 1.1.14) we can find some indiscernible sequence I = hci | i ∈ ωi of |x|- tuples
satisfying theEM- type ofA. Note that for any two disjoint finite sets I0 andI1 ofI the partial type {φ(c, y) | c∈I0} ∪ {¬(φ(c, y)) | c∈I1} is consistent. In particular the type Γ(y) ={φ(a2i;b)∧ ¬φ(a2i+1;b) :i < ω}is finitely consistent, and by compactness there is
an element b ∈ Ml realizing Γ(y). So, M |=φ(c
i;b) if and only ifi is even.
Lemma 1.1.23. T is a dependent theory if and only if for any indiscernible sequence
hai | i ∈ ωi, and every formula φ(x) (possibly with parameters) there is an i ∈ ω such
that either M |=φ(aj) for all j > i, or M |=¬(φ(aj)) for all j > i.
Proof. (⇒) Assume thatT is a dependent theory and letI =hai |i∈ωian indiscernible
sequence and φ(x, b) a formula and b ∈ Ml. If the conclusion does not hold, then for
every i < ω there are i≤j0 < j1 such that M |=φ(aj0;b)∧ ¬φ(aj1;b).
This allows us to consider a subsequence S =haij :j < ωisuch that M |=φ(aij;b) if
put ai0 =aj0, ai1 =aj1. Now, apply the same for i=i1 to find elements ai2, ai3 such that
φ(ai2, b)∧ ¬φ(ai3, b). Continuing with this construction we obtain the desired sequence.
SinceS is an increasing subsequence of B, it is also indiscernible. By Theorem 1.1.22, we conclude that φ(x, y) has the independence property. ThusT is not a dependent theory. (⇐) Assume that there is a formula φ(x, y) that has the independence property. By Theorem 1.1.22, there is an indiscernible sequence hai | i ∈ ωi and a tuple b such that
M |=φ(ai, b) if and only if iis even. Then, it is impossible to find an element i∈ω, such
that either M |=φ(aj, b) for all j > i, or M |=¬(φ(aj, b)) for allj > i.
Remark 1.1.24. We can say that φ(x) has an average value in the sequence hai :i∈ωi
if there is i ∈ω such that M |=φ(aj)↔φ(aj+1) for all j > i, which is the conclusion of
the previous lemma.
Thus, Lemma 1.1.23 can be stated as follows: T is a dependent theory if and only if every formula φ(x)∈ L(M) has average value in every indiscernible sequence.
Given an indiscernible sequence I = hai : i < ωi in a dependent theory, the set of
formulas φ(x;b) that holds cofinally in I is a complete type, known as the average type of the sequence I.
Corollary 1.1.25. Suppose that T is a dependent theory. Let φ(x, y) be an L-formula and hbi | i ∈ ωi an indiscernible sequence. Then, {φ(x, b2j)△φ(x, b2j+1) | j ∈ ω} is
inconsistent.
Proof. Assume by contradiction that it is consistent, since M is saturated, there is an element c such that for all j ∈ ω, M |= φ(c, b2j)△φ(c, b2j+1). This contradicts Lemma
1.1.23, since the formula φ(c;y) will not have average value in hbi | i∈ωi.
1.2
Groups and measures in dependent theories
In [7] Hrushovski, Peterzil and Pillay stated several ideas relating measures and the de-pendence property, in order to prove that any definably compact group in an o-minimal
theory has the fsg property. However, in order to understand their statements we will need to explain several tools, such as measures, Keisler measures and Keisler invariant measures, which are explained in Section 1.2.1, based on the book [10] of P. Simon, who did a very detailed explanation of these topics on Chapter 7.
Secondly, we present a detailed exposition of the main propositions of [7], in particular we recall those results that establish a relationship between the existence of an invariant measure and some properties of the dependent theories. Since those statements are going to be relevant to show that any definable bounded group in Presburger Arithmetic theory have the fsg property, we explain all the theory developed by Hrushovski, Peterzil and Pillay in the Section 1.2.3.
Additionally, in Subsection 1.1.3 we define the fsg property for definable groups in dependent theories, and we recall some relevant statements developed in [7], in order to characterize G00 as the stabilizer of a global generic type.
In the following statements we will be working with T a complete dependent theory, and Mwill indicate a very saturated model of T.
1.2.1
Measures and Keisler Measures
Definition 1.2.1. Let X be a set. A collection Ω of subsets of X is called a boolean algebra on X if ∅, X ∈ Ω, and Ω is closed under complements and finite unions. If in addition Ω is closed under countable unions, we say that it is a σ-algebra.
Example 1.2.2. LetX be a definable set. Then the following sets are boolean algebras onX:
• the set DefA(X) ={Y ⊆X | YisA-definable}. We denote simply asDef(X) the
set of the definable subsets of X in M.
Definition 1.2.3 (Measures). Let X be a set, and Ω be a boolean algebra on X.
1. Afinitely additive measure onX(with respect to Ω) if a functionµ: Ω→R≥0∪{∞}
satisfying the following:
(a) µ(∅) = 0
(b) If A1,· · · , An are pairwise disjoint elements in Ω then
µ
n [
i=1 Ai
!
=
n X
i=1
µ(Ai).
2. If in additionµ(X) = 1, we say that µis a probability measure on X.
3. A finitely additive measure is calledσ-additiveif Ω is aσ-algebra and for every count-able family {Ai :i∈N}of disjoint subsets in Ω we have µ
∞ [
i=1 Ai
!
=
∞ X
i=1
µ(Ai).
Example 1.2.4. The following are standard examples in measure theory:
• LetX =Nand defineµ:P(N)→R≥0∪{∞}asµ(A) =|A|. Then,µis aσ-additive measure. (µis called the counting measure)
• Let X be a non-empty set, and fix an element p ∈ X. Then the function µp :
P(X)→ {0,1} defined as :
µp(A) =
1 if p∈A 0 otherwise
is a σ additive measure. This measure is known as the measure centered in p.
Now we move on to describe the so-called Keisler measures. For a given definable sets X inM, we indicate with DefA(X) the collection of all definable subsets of X with
parameters over A. We denote by Def(X) the set DefM(X). Also, we refer the reader to
[10] for more details.
Definition 1.2.5 (Keisler measures).
1. A Keisler measure µ on X over A is a probability measure defined on the Boolean algebra DefA(X).
2. A global Keisler measure is a probability measure defined on Def(X).
3. If µ is a Keisler measure on DefB(X) and A ⊆B, we write µ↾A for the restriction
of µto DefA(X).
Let φ(x, y) be an L-formula with x a tuple of variables with sort X. Given a Keisler measureµon Def(X), we can define the functionµφ:M|y| →[0,1] byµφ(a) :=µ(φ(x, a)).
Conversely, a global Keisler measure on X is completely determined by the family of functions {µφ |φ =φ(x, y) is anL-formula } and µφ :Sφ →[0,1].
Theorem 1.2.6. [Extension of measures] Let µ a Keisler measure on X over a model
M0, then µ can be extended to a global Keisler measure.
Proof. Letµa Keisler measure onXover a modelM0, viewed as a map from the definable subsets of X(M0) over M0 to [0,1]. Let L′ be the two-sorted language L ∪ {M,(I,+, < ,0,1), µφ} where the sort M carries the L-structure, I is a sort of an ordered interval
with end points 0 and 1, and µφ are functions from M|y|→ I. Consider the L′-structure
A=hM0,[0,1],+, <, µφi where:
• M0 is the original structure adding a constant for each element,
• [0,1] is the real interval, with the order and the addition modulo 1,
Let A′ = hM′,[0,1]′,+, <, µ′
φi be a saturated model of A. Let st: [0,1]′ → [0,1] be
the standart part map. We will show that the map st◦µ′ : Def(X) → [0,1] is a global Keisler measure that extends µ.
Firstly, note that any q ∈ Q ∩[0,1] is L′-definable. Let m ⊂ M0. Since A′ is an
elementary extension of A, A |= µ(φ(x, m)) ≤ q if and only if A′ |= µ(φ(x, m)) ≤ q.
Therefore, µ′
φ(m) ∈ [0,1]′ is an element in the same rational cut as µ(φ(x, m)), and by
composing with the standard map we have that st(µ′
φ(m)) :=µ′(φ(x, m)) = µ(φ(x, m)).
Therefore, µ′ ↾M0=µ.
We show now that µ′ is a probability measure. Note that µ′(∅) = µ(∅) = 0 and µ′(X) =µ(X) = 1, since this is a first order property. Let φ(x, y1) and ψ(x, y2) be two
M-formulas. Again since A′ is an elementary extension of A we have
A |=∀y1∀y2(∀z(¬φ(z, y1)∧φ(z, y2))→(µφ(y1) +µψ(y2) =µφ∨ψ(y1, y2))
⇔A′ |=∀y1∀y2(∀z(¬φ(z, y1)∧φ(z, y2))→(µφ(y1) +µψ(y2) = µφ∨ψ(y1, y2)) that is, µ′ : Def(X)→ [0,1]′ is finitely additive and since the standard map is linear, we
have that st◦µ′ is a finitely additive measure.
Lemma 1.2.7. Let B = hbi | i ∈ ωi be an indiscernible sequence, then we can find a
sequence C =hci | i∈ωi such that:
• C satisfies the EM-type of B,
• C is µ- indiscernible.
Proof. It follows by the Erdos-Rado theorem and compactness. For a detailed explanation we refer the reader to [15] Lemma 5.13.
Lemma 1.2.8. Let µ be a Keisler measure on X. Let x be a variable of sort X. Let
ǫ >0, µ(φ(x, bi))≥ǫ for all i∈ω. Then, F ={φ(x, bi) | i∈ω} is consistent.
Proof. Consider the extended structure A = hM,[0,1], µφi induced by the Keisler
mea-sureµonX. Notice that by compactness it is enough to show thatF is finitely consistent, and this condition is preserved by the EM-type of hbi :i∈ωi.
By Lemma 1.2.7 we may assume thatB =hbi |i∈ωiisL′-indiscernible. In particular,
B is µ-indiscernible.
LetYbi the set inMdefined by φ(x, bi). By µ-indiscernibility, whenever i1 <· · ·< in,
and j1 < · · · < jn we have : µ
Ybi1 ∩ · · · ∩Ybin
= µYbj1 ∩ · · · ∩Ybjn
= rn for a fixed
rn∈[0,1]. We have by hypothesis that r1 ≥ǫ >0.
Suppose that there is n ∈ ω such that F is n-inconsistent. Then
n \
i=1
Ybi =∅. Choose a
maximal k ∈ N, such that rk > 0 and for each j ≥ 0 consider the set Zj = Yb1 ∩ · · · ∩
Ybk−1 ∩ Ybk+j. We have by µ-indiscernibility that µ(Zj) = rk and by maximality of k,
µ(Zj1 ∩Zj2) = µ(Yb1 ∩ · · · ∩Ybk−1 ∩Ybk ∩Ybk+1) = 0 for every 0 ≤ j1 < j2. Let M ∈ N
be such that M ·rk > 1, then µ(X) ≥ µ SMj=1Zj
= Pnj=1µ(Zj) = M · rk > 1, a
contradiction.
We conclude that F ={φ(x, bi)| i∈ω}is consistent.
Definition 1.2.9 (∼µ). Let X be a definable set and µ a Keisler measure over X. Let
Y, Z two definable subsets of X we define Y ∼µZ if and only if µ(Y△Z) = 0.
Lemma 1.2.10. ∼µ is an equivalence relation inDef(X).
Proof. Since µ(X△X) = µ(∅) = 0, we have that ∼µ is reflexive. Symmetry comes from
the fact thatX△Y =Y△X. Finally, assume X ∼µY and Y ∼µZ, and notice that since
X△Z = (X△Y)△(Y△Z), we have that µ(X△Z) ≤ µ(X△Y) +µ(Y△Z) = 0 + 0 = 0. We conclude thatX ∼µ Z.
1.2.2
Keisler measures in dependent theories
In this section we state some theorems of [7], which establish a relationship between the dependence property and the Keisler Measure. This connection will be a crucial point to show that any definable bounded group in Presburger Arithmetic have the fsg property.
Lemma 1.2.11. Assume that T is a dependent theory. Let µbe a global Keisler measure
onX,φ(x, y)a formula withxof sortX andǫ >0. Then, there is no sequencehbi |i∈ωi
such that for all i6=j, µ φ(x, bi)△φ(x, bj)
≥ǫ.
Proof. Assume for a contradiction that there is such sequence hbi | i ∈ ωi. By Lemma
1.2.7, we may assume that hbi | i ∈ ωi is an indiscernible sequence. Thus, the sequence
hbibi+1 : i < ωi is also indiscernible and since µ(φ(x;bi)△φ(x;bi+1)) ≥ǫ for every i < ω, then by Lemma 1.2.8 we have that the set {φ(x, b2j)△φ(x, b2j+1)} must be consistent.
However, this contradicts that T is dependent by Corollary 1.1.25.
Corollary 1.2.12. Assume that T is a dependent theory and let µ be a global Keisler
measure on a definable setX. Then there are only boundedly many∼µ-classes of definable
subsets of X.
In particular there is a small model M0, such that every definable subset Y of X is
∼µ-equivalent to someM0-definable subset X.
Proof. Assume for contradiction that there are unboundedly many∼µ- classes of definable
subsets of X.
Claim 1: There is a formula ψ(x, y) and a sequence hbi | i∈ωi such that for all i6=j
µψ(x, bi)△ψ(x, bj)
>0
Let C = {[φ(x, b)]∼µ | b ⊆ M, φ(x;y) ∈ L} be the collection of all ∼µ-classes, and
consider the function f : L × P<ω(M) → C given by f(φ(x, y);b) 7→ [φ(x;b)]
a surjective function, and since C is unbounded, κ = |C| > |T|. By pigeonhole principle there is a formulaψ(x, y) such that the collection{[ψ(x;b)]∼µ :b ⊆ M}contains infinitely
many ∼µ-classes. Then, we can choose an infinite sequence hbi | i∈ ωi such that for all
i6=j, µψ(x, bi)△ψ(x, bj)
>0. Claim 1. By Lemma 1.2.7, there is a µ-indiscernible sequence hdi :i < ωiwith the same
µ-EM-type of hbi : i < ωi. Observe that by µ-indiscernibility, µ ψ(x, di)△ψ(x, dj)
= ǫ >0 is constant for every i < j < ω. This contradicts Lemma 1.1.25, and we conclude that the
∼µ-classes of the definable subsets of X are bounded.
Consider now a set H = {φα(x, bα) | α ∈ λ} of representatives for the equivalence
relation ∼µ. Since there are only boundedly many ∼µ-classes, we may assume that λ <
|M|. Let A = Sα<λbα, and notice that |A| ≤ λ ·ω = λ. By downward
L¨owenheim-Skolem, there is a model M0 ≺ M of size λ containing A. Notice that M0 is a small model, and by construction, for any definable subset Y of X, there is an M0-definable set Z, such thatY ∼µZ.
1.2.3
Groups with fsg
In this section we recall several results presented in [7]. In this paper Hrushovski, Peterzil and Pillay presented a property called the fsg (finitely satisfiable generics), and they use this property to characterize the subgroup G00 as the stabilizer of a generic global type.
Definition 1.2.13. 1. Let (G,·, e) be a definable group. We say that a type p(x) is
centered in G if p(x)|= (x∈G).
2. Given g ∈G and a typep(x) centered in G, we define theleft-translate of p by g to be the typeg·p:={φ(x) :φ(g−1x)∈p(x)}.
Notice that ifp(M) is the set of realizations ofp(x), theng·p(M) corresponds with the set of realizations of the type g·p.
Definition 1.2.14 (Generics). LetX be a definable subset of a definable group (G,·, e), and p(x) a type centered in G.
1. We say that X is left- generic (resp. right-generic) if finitely many left- translates (resp. right-translates) of X cover G. Namely, there are g1,· · · , gk ∈ G such that
G=
k [
i=1 gi·X.
2. If X is both left-generic and right-generic, we will say that X is generic.
3. We say that p(x) is a generic type if all its formulas are generic.
Example 1.2.15. The following are examples of generic sets in definable groups: Let T be any theory and G be a definable group of T. Clearly, X =G is generic.
Let T be an extension of the o-minimal theory T h(R,+,·, <,0,1) and let M be a very saturated model of T. Any cofinite subset of a definable group G in T is generic.For example, consider the definable group S1 = {(x, y) | x2 +y2 = 1}, the set S1\{1} is generic.
Definition 1.2.16. [A group with fsg] Let (G,·, eG) be a definable group. We say G has
finitely satisfiable generics (fsg) if there exists a small elementary submodel M0 and a global generic type p(x)∈S(M), such that p(x)|=x∈G, and for every g ∈G(M), the left-translate g·p(x) :={φ(x) | φ(g−1·x)∈p(x)} of p(x) is finitely satisfiable in M0.
Fact 1.2.17. Let p(x) be a complete global type centered in G and assume that X is
generic. Then there is some g ∈G such that X∈g·p.
Proof. SinceX is generic, there areg1,· · · , gk∈Gsuch that G= k [
i=1
gi·X
. Letφ(x, b) be a formula which defines G, and ψ(y, a) the formula which defines X. Notice that:
M |=∀xφ(x, b)↔
k _
i=1
∃y ψ(y, a)∧x=gi·y
Since φ(x, m) ∈ p(x), then
k _
i=1
gi·X ∈p, and by completeness of p there is some i ≤ k,
such thatgi·X ∈p. Therefore X ∈gi−1·p(x).
Lemma 1.2.18. Assume that G has fsg, witnessed byp(x) andM0. LetX be a definable
subset of G, then:
1. X is left-generic if and only if X is right-generic, then we say simply that X is
generic.
2. X is generic if and only if every left (right)-translate of X meets M0.
3. The type p(x) is generic, as well as any other right or left translate of p(x).
4. If X is generic and X =X1∪X2, where X1 and X2 are definable, then one of them is generic.
Proof. 1. Assume that X is left-generic, then every translate ofX is also left-generic. Letc be an arbitrary element of G. By Fact 1.2.17, there is some g ∈G, such that c·X ∈g·p(x). By fsg, c·X∩G(M0)6=∅. Letb ∈ c·X)∩G(M0), then there is somex∈X, such thatb =c·x, and soc−1 =x·b−1 ∈X·b−1. Since b−1 ∈G(M
0), we have shown that for any element c ∈ G, there is an element b ∈ G(M0) such that c∈X·b.
If X were not right-generic, then for any h1,· · · , hn ∈ G(M0) there is an element
c∈ G such that c /∈
n [
i=1
X·hi. Let φ(x, m) and ψ(x, a) be the formulas defining G
and X, respectively. Consider the type σ(z) = {φ(z, m)} ∪ {∀x ψ(x, m) → ¬(c = x·h) | h ∈ G(M0)}. Notice Σ(c) is finitely consistent because X is not right generic. By saturation of M there is a tuple c such that M |= Σ(c), then c ∈ G, and ∀h∈G(M0),b /∈X·h, which contradicts the result obtaining in the previous paragraph. We conclude then that X is right generic. The other direction follows in the same way.
2. It follows directly from the previous argument.
3. By hypothesis, for every X ∈ p(x), and every element g ∈ G, we have g ·X ∩
G(M0) 6= ∅. By (2), X is generic. Since every formula in p(x) is generic, we conclude that p(x) is a generic type.
4. Let X1, X2 two definable subsets of G and assume that X = X1 ∪X2 is generic. By Fact 1.2.17, there is an element g ∈G such that X ∈g·p(x), and since p(x) is complete, either X1 ∈g·p(x) or X2 ∈g·p(x). Thus, either X1 orX2 is generic.
Remark 1.2.19. Notice that (4) implies that the collection IG = {Y ⊂ G | Y is
non-generic in G} is an ideal in the algebra of the definable subsets ofG.
Definition 1.2.20 (∼I). Let (G,·, e) be a definable group. We define the relation ∼I on
Def(G) given by X∼I Y if and only if X△Y is non-generic.
Lemma 1.2.21. If IG is an ideal, then the relation ∼I is an equivalence relation.
Proof. Note that X△X =∅, which is non-generic. So, ∼I is reflexive. Symmetry follows
from the fact that X△Y =Y△X.
To show transitivity, suppose that X ∼I Y and Y ∼I Z, then X△Y and Y△Z is
non-generic. Since X△Z ⊆(X△Y)∪(Y△Z) and both (X△Y),(Y△Z) are non-generic, the fact that IG is an ideal implies that X△Z is non-generic. Thus, X∼I Z.
Definition 1.2.22 (Subgroup of bounded index). LetG be a definable group in Mand let H be a subgroup of G. We say that it has bounded index if [G(M) : H] = α < κ, where κ=|M|. (In particular |M| ≥2|T|)
Definition 1.2.23. Let (G,·, e) be a definable group. We say thatGisdefinably amenable
if there exists a Keisler measure on Def(G) that is G-invariant.
Lemma 1.2.24. Suppose that T is a dependent theory and let G be a group definable in
1. There is only a bounded number of definable subsets of G modulo the equivalence
relation X ∼I Y.
2. For each definable generic subset X ⊂ G, the stabilizer of X in G, defined as:
StabI(X) := {g ∈G | g·X△X is non-generic}
is a type definable subgroup of bounded index.
Proof. 1. Observe that ifXis generic thenµ(X)>0. Indeed, ifg1,· · · , gkare elements
niG such that G=
k [
i=1
gi·X, then
1 = µ(G) =µ
k [
i=1 gi·X
≤
k X
i=1
µ gi·X
=
k X
i=1
µ X=kµ(X), and we have µ(X) ≥ 1
k > 0. So, given two definable subsets Y, Z ⊆ X such that
Y 6∼I Z, we have that Y△Z is generic, and µ(Y△Z) >0. Thus, Y 6∼µ Z. Hence,
if there are unboundedly many ∼I-classes, there are also unboundedly many ∼µ
-classes, and this contradicts Lemma 1.2.12.
2. Let X be a generic set. We will see that StabI(X) = {g ∈ G | g ·X△X is
non-generic} is a type definable subgroup of G of bounded index.
First we show that StabI(X) is a subgroup of G. Notice that e ∈ StabI(X) since
e·X△X =X△X=∅, which is non-generic. Also, ifg ∈StabI(X), then g·X
△X is non-generic, and g−1·X△X =g−1 X△ g·X is also non-generic because it is the translate of a non-generic set. So, g−1 ∈Stab
I(X).
Suppose now that g1, g2 ∈ StabI(X), we want to show that g2 · g1 ·X
△X is non-generic. Observe that:
g2·g1·X△X =g2· (g1·X)△(g2−1·X)
=g2· (g1·X)△X△ (g2−1·X)△X
X)△Xare non-generic, because they are subsets of the non-generic sets (g1·X△X) and (g2·X△X), respectively. Since IG is an ideal, we conclude thatY1∪Y2 is
non-generic, and finally g2·(Y1∪Y2) = g2·g1·X△X is also non-generic.
We show now that StabI(X) is type definable. For eachn ∈ωconsider the formula
φ(g) =∀h1· · ·hn n ^
i=1
(hi ∈G)→ ∃c c∈G∧ n ^
i=1
c /∈(hi·(g·X△X)) !!!
and letAnbe the set of points inMdefined byφ(g). Observe that StabI(X) = \
n∈ω
An,
therefore it is a type-definable set.
We finally show that StabI(X) has bounded index inG. Observe thatg1·StabI(X)6=
g2 ·StabI(X), if and only if g2−1·g1 ∈/ StabI(X) if and only if g2−1·g1·X
△X is generic, which implies g1·X
△ g2·X
is generic, and thus g1 ·X
6∼I g2·X
. Since, there are only boundedly many ∼I- classes, then there is also a bounded
number of cosets of StabI(X) in G.
Definition 1.2.25. LetA be a small set of parameters and G be a definable group. We define G00
A as the smallest type A-definable subgroup of bounded index of G.
Remark 1.2.26. Shelah proved that if G is definable in a dependent theory, then for every set of parameters A we have that G00
A = G00∅ . Then, since Presburger Arithmetic
is a dependent theory, we will refer to the smallest type-definable subgroup of G just as G00.
Definition 1.2.27 (Stabilizer of a type). Letp be a global type, then
Stab(p) ={g ∈G |g·p=p}
Remark 1.2.28. For every global type p, we have that Stab(p) is a subgroup ofG.
1. There is a bounded of number of global generic types.
2. G00 exists.
3. For each global generic type q, G00 = \
X∈q
StabI(X) =Stab(p).
Proof. 1. Let GGT be the set of the generic global types centered in G, and consider the function S : GGT → P(P(M0)), defined as S(p) = {X ∩G(M0) | X ∈ p}. Since |P(P(M0))|= 22
M0
is bounded, it is enough to show that S is injective. Letp, qbe two different generic global complete types, and take a formula φ(x)∈p such that ¬(φ(x)) ∈ q. Let A = φ(M)∩G(M0) and notice that A ∈ S(p), we will show that A /∈ S(q). Suppose by contradiction, that A ∈ S(q). Then there is a formula ψ(x) ∈ q, such that ψ(M)∩G(M0) = A = φ(M)∩G(M0), but then (ψ(M)∩ ¬φ(M))∩G(M0) = ∅. So, the set defined by the formula ¬φ(x)∧ψ(x) inMdoes not meet M0, and by Lemma 5.3(2) this implies that the set defined by
¬φ(x)∧ψ(x) is not generic, contradicting thatq is generic. Therefore, the function S is injective, and |GGT | ≤22|M0|
. We conclude that there are only boundedly many global generic types.
2. Using the notation from (1), let |GGT | = λ be the (bounded) cardinality of the collection of global generic types. Let H be a type-definable subgroup of G with bounded index inG and let pbe some global fixed generic type.
Claim: Each coset of H is in some translate ofp.
Assume thatH is defined by ^
i<α
φi(x, ai)
, where eachai is a tuple fromM. Since
H has bounded index in G, |{g·H | g ∈G}|=γ < |M|. Let {gj ·H}j<γ be a set
of representatives ofG/H.
For a contradiction, let us assume that for all gj ∈ G, gj ·H /∈ p. Then, for every
¬ gj·φij(x, aij)
∈p.
Consider the type Γ(y) ={y∈G} ∪ {¬ gj ·φij(x, aij)
|j < γ}, which is consistent since it is contained in p. By saturation, there is an element b ∈ Mrealizing Γ(y), that is, an element b such that b∈G and
b /∈ [
j<γ
gj·φij(x, aij)
⊇ [
j<γ
gj·H =G,
which is clearly a contradiction. Claim. Now, by the claim above, we have |G/H|< λ, since for every g ∈G, g·p is also a generic global type. Since this bound is independent from the monster model, we conclude that G00 exists. (In fact, let
H={H ⊆G | H is type definable and |G/H| ≤λ}, and notice that G00= \
H∈H
H. Since this intersection is a bounded intersection of
subgroups of bounded index it also has bounded index in G. It means that G00 exists).
3. Fix a gobal generic type q, such that q |= x∈ G. Notice that for every definable set X ∈ q, X is generic, then by Lemma 1.2.24 (2), G00 ⊆ Stab
I(X). Therefore
G00⊆T
X∈qStabI(X).
Claim 1: Let X ∈q and g ∈StabI(X), then g·X ∈q
Let X ∈ q and g ∈ StabI(X), then g ·X
△X is non-generic. Since X is generic and X = X\g·X∪ X ∩(g ·X), then X ∩g ·X is generic (because IG is an
ideal and X\(g·X) is non-generic). Additionally, since X ∈q, by completeness, then X\g·X∈q orX∩(g·X)∈q, then X∩(g ·X)∈q, therefore g·X ∈q.
Claim 2: \
X∈q
StabI(X)⊆Stab(q)
Let g ∈ \
X∈q
StabI(X), then by the step 1, we know that for every definable set
X ∈q, g·X ∈q. Then, g ∈Stab(q).
Claim 3: Stab(q)⊆G00
Fix an element h∈ Stab(q). Since G00 has bounded index, then there is some ele-ment g ∈G, such that (g·q)⊆G00, it means that the translate C =g−1·G00 is in q. Therefore, h·g−1 ·G00 = g−1 ·G00, which implies that h ∈ g−1·G00·g ⊆ G00 (because it is a normal subgroup), then h∈G00 as we desired.
Summarizing we have thatG00⊆ \
X∈q
StabI(X)⊆Stab(q)⊆G00, therefore all these
sets are equal.
1.3
Amenable Groups
The notion of definably amenable group presented in the subsection Section 1.2.3 can be strengthened to a property called amenability. Amenable groups have been a topic of study during a long time.There are some relevant theorems in this area, such as ev-ery abelian group is amenable, and moreover that evev-ery abelian by finite group is also amenable. These results will play a key role in our study of the definable groups in Pres-burger Arithmetic, and this is for this reason why we present a detailed explanation of these topics in this section. The theory of amenable groups can be found by the reader in [11].
Definition 1.3.1 (Amenable Group). Let (G,·, e) be a group. We will say that G is
1. The function µis a finitely additive measure,
2. The functionµisG-left-invariant, that is, for allg ∈G, andA ⊆G,µ(A) =µ(g·A), 3. µ(G) = 1.
Lemma 1.3.2. The following statements hold:
1. Every finite group is amenable.
2. If G is an amenable group, and H is a subgroup of G then H is also amenable.
3. If G is an amenable group and N is a normal subgroup of G, then G/N is also
amenable.
4. LetGbe a group andN a normal subgroup ofGsuch thatN andG/N are amenable.
Then G is amenable.
Proof. 1. Let (G,·, e) be a finite group. Assume that |G|=n, then define µ:P(G)→
[0,1] as µ(A) = |A|n .
• µ is a finitely additive measure: Notice that µ(∅) = 0n = 0, and if A, B ⊂ G are disjoint, then|A∪B|=|A|+|B|. So, µ(A∪B) = |A∪B|
n =
|A|+|B|
n =
µ(A) +µ(B).
• µis G-invariant: Let g ∈Gand consider fg :G→G defined as fg(x) =g·x.
Note that fg is injective: if fg(x1) = fg(x2) then g ·x1 = g·x2 which implies thatx1 =x2 by left cancelation. Thus, for allA⊂Gwe have |g·A|=|{fg(a) :
a∈A}|=|A|, and thereforeµ(g·A) =µ(A).
• µ(G) = nn = 1.
We conclude that Gis amenable.
2. Let µ be the finitely additive measure witnessing that G is amenable. Let M be a set of representatives for the collection of the right cosets of H inG. Consider the function ν :P(H)→[0,1] defined by ν(A) =µ[{A·g | g ∈M}.
• ν is a finitely additive measure: Notice that ν(∅) = µ(∅) = 0 and ν(H) = µ S{H·g | g ∈M} =µ(G) = 1. Consider now A, B disjoint subsets of H. If g1, g2 ∈M and g1 6=g2 then g1·A∩g2·B =∅, since they are contained in different cosets. Therefore, it is enough to show that g·A∩g·B =∅.
If g·A∩g ·B 6= ∅, then there exist a ∈ A and b ∈ B such that g ·a = g ·b, and we can conclude that a =b, which contradicts the fact that A and B are disjoint.
Hence, [{A·g | g ∈M} ∩[{B·g | g ∈M} =∅, and we have ν(A∪B) = µ [{(A∪B)·g | g ∈M}
=µ [{(A·g ∪B ·g | g ∈M}
=µ [{(A·g | g ∈M} ∪[{(B·g | g ∈M}
=µ( [{(A·g | g ∈M}) +µ( [{(B·g |g ∈ M}) =ν(A) +ν(B)
• ν isH- invariant: Leth∈H and A⊆H. Then,
ν(h·A) = µ [{((h·A)·g |g ∈M}=µ [{h·(A·g) |g ∈M}
=µ h·[{(A·g) | g ∈M} =µ [{(A·g) | g ∈M}
=ν(A).
Thus H is amenable.
3. Let µ be a measure witnessing that G amenable and N a normal subgroup of G. Consider ν :P(G/N)→[0,1] defined as ν(A) =µ(SA).
• ν is a finitely additive measure on G/N:
Notice that ν(G/N) = µ(SG/N) = µ(G) = 1 and ν(∅) = µ(∅) = 0. Let now A, B be disjoint subsets of G/N. We claim that SA∩SB =∅.
If z ∈ SA∩SB then there exist g1, g2 ∈ G and n1, n2 ∈ N such that z = g1·n1 ∈ SA and z =g2·n2 ∈SB. So g2 = g1·n for n =n1·n−21 ∈ N and g1·N =g2·N ∈A∪B contradicting that A∩B =∅).
Then ν(A ∪B) = µ S(A∪ B) = µ SA∪ SB = µ(SA) + µ(SB) = ν(A) +ν(B), as desired.
• ν is G/N invariant:
Let g·N ∈ G/N and A ⊆ G/N, say A ={gi·N : i ∈ I}. Then, (g ·N)A =
{(g·gi)N :i∈I}and hence
ν((g·N)A) =µ [
i∈I
(g·gi)N
=µ {g·gi·n |n∈N, i∈I}
=µ g· {gi·n |n ∈N, i∈I}
=µ {gi·n |n∈N, i∈I}
=µ [
i∈I
gi·N
=µ [A =ν(A).
Therefore, G/N is amenable.
4. Letν1 andν2 be the left-invariant measures ofN and G/N respectively, witnessing amenability of these groups. For any A ⊆ G, consider fA : G → R≥0 defined by
fA(g) =v1(N∩g−1·A). Assume thatg1·N =g2·N, then there is ann∈N such that
g2−1·g1 =n. Then g2−1=n·g1−1, andfA(g2) =ν1(N∩g−21·A) =ν1(N∩n·g1−1·A) = ν1(n·(N ∩g1−1 ·A)) = ν1(N ∩g1−1 ·A) = fA(g1). Then fA may be regarded as a
bounded real-valued function with domain G/N, consider ˆfA : G/N → R defined
as ˆfA(g·N) =fA(g). Finally, define µ(A) = Z
G/N
ˆ fAdν2.
• µ is a finitely additive measure: We have µ(G) =
Z
G/N
ˆ fGdν2 =
Z
G/N
1dν2 =ν2(G/N) = 1
and µ(∅) =
Z
G/N
ˆ
f∅ dν2 = Z
G/N
Consider now A, B ⊆ G such that A∩B = ∅. Notice that ˆfA∪B = ˆfA+ ˆfB,
since for all g ∈G, ˆ
fA∪B(g) =fA∪B(g) =ν1(N ∩g−1·(A∪B)) =ν1((N ∩g−1·A)∪(N ∩g−1·B)) =ν1(N ∩g−1·A) +ν1(N ∩g−1·B) =fA(g) +fB(g) = ˆfA(g) + ˆfB(g).
Therefore,
µ(A∪B) =
Z
G/N
ˆ
fA∪Bdν2 =
Z
G/N
ˆ
fA+ ˆfBdν2
=
Z
G/N
ˆ fAdν2+
Z
G/N
ˆ
fBdν2 =µ(A) +µ(B),
as desired.
• µ is G-invariant: Let g ∈G and A⊂G, notice that: ˆ
fg·A(h·N) =fg·A(h) =ν1(N∩h−1 ·g ·A)
=ν1(N ∩(g−1·h)−1·A) =fA(g−1·h)
= ˆfA(g−1·h·N)
This implies that
Z
G/N
ˆ
fg·Adν2 = Z
G/N
ˆ
fAdν2. In fact, for any finite collection
{B1, . . . , Bn} of disjoint subsets of G, n
X
i=1
ciχBi ≤fˆA(g
−1·) if and only if
n X
i=1
So, since the integrals are approximated by simple functions, we have Z G/N n X i=1
ciχBidν2 =
n X i=1 ci Z G/N
χBidν2
=
n X
i=1
ciν2(Bi) = n X
i=1
ciν2(g·Bi)
= Z G/N n X i=1
ciχg·Bidν2,
and we can conclude that:
Z
G/N
ˆ
fA(g−1·t)dν2 =
Z
G/N
ˆ fA(t)dν2
Then, G is amenable.
Proposition 1.3.3. If G is a direct union of a directed system of amenable groups
{Gα}α∈I then G is amenable.
Proof. Assume that G= [
α∈I
Gα, where each Gα is amenable witnessed by the measure
µα. Consider the topological space [0,1]P(G) with the product topology, and let Mα be
the collection of all functions µ∈ [0,1]P(G) such that µ is a finitely additive measure on G, µ(G) = 1 and µ(g·A) =µ(A) for all g ∈Gα.
- Claim 1: Mα is not empty
Define µ : P(G) → [0,1] as µ(A) = µα(A∩ Gα). Notice that µ(∅) = µα(∅) = 0 and
µ(G) =µα(G∩Gα) =µα(Gα) = 1. Also, if A, B ⊆Gare disjoint, then
µ(A∪B) =µ((A∪B)∩Gα) =µα((A∩Gα)∪(B∩Gα))
Finally, let g ∈Gα and A⊆G. Then
µ(g·A) =µα(g·A∩Gα) =µα(g·(A∩Gα)) = µα(A∩Gα) =µ(A).
Therefore, µ∈ Mα. Claim 1
- Claim 2: Mα is closed
We will show thatMc
αis open. Suppose thatf 6∈ Mα. Then eitherf(∅)6= 0, orf(G)6= 1,
or f(A) 6= f(g ·A) for some g ∈ Gα and A ⊆ G, or f(A∪B) 6= f(A) +f(B) for some
disjoint subsets A, B of G. Let us write f(A) = fA ∈ [0,1] for A ⊆G. We can consider
the four cases separately:
• If f∅ 6= 0, consider the set V =V0 = Y
X∈P(G)
VX0 where V0
∅ = (0,1] and VX0 = [0,1]
for X 6=∅.
• IffG 6= 1, take V =V1 = Y
X∈P(G)
VX1 where V1,G = [0,1) andVX1 = [0,1] for X 6=G
• Iffg·A6=fA for some g ∈Gα and A⊆G, we may assume without loss of generality
that fg·A < r < fA for some r ∈ [0,1] and take V =Vg,A = Y
X∈P(G)
VXg,A where Vg·Ag,A= [0, r),VAg,A = (r,1] and V1,X = [0,1] for X 6=g·A, A. Notice that iff′ ∈V,
then fA′ < r < fg·A′ .
• IffA+fB6=fA∪B for aA, B ⊆G disjoint, we can take
ǫ= |fA+fB−fA∪B| 4 >0
and consider the setV =VA,B = Y
X∈P(G)
VXA,BwhereVAA,B = (fA−ǫ, fA+ǫ),VBA,B =
Notice that if f′ ∈V, then
|fA′ +fB′ −fA∪B′ |=|fA∪B′ −fA′ −fB′ |
=|(fA+fB−fA∪B)−(fA′ −fA+fB′ −fB−fA∪B′ −fA∪B)|
≥ |(fA+fB−fA∪B)| − |fA′ −fA+fB′ −fB−fA∪B′ −fA∪B|
≥ |(fA+fB−fA∪B)| − |fA′ −fA| − |fB′ −fB| − |fA∪B′ −fA∪B|
≥4ǫ−ǫ−ǫ−ǫ=ǫ >0. and so, f′
A+fB′ 6=fA∪B′ .
In any case, we have that V is an basic open [0,1]P(G) with the product topology, f ∈V, and V ⊆ Mc
α because given f′ ∈ V, one of the conditions that define Mα fails for f′.
Therefore, we can conclude that Mα is closed. Claim 2
- Claim 3: {Mα}α∈I has the finite intersection property.
Since G is the direct limit of a directed system, given α1, . . . , αn∈ I there is γ ∈I such
that Gγ embeds the groups Gα1, . . . , Gαn. We may assume without loss of generality
that each Gαi is a subgroup of Gγ. Notice that if a probability measure that is Gγ
–left-invariant, then it is Gαi-left-invariant for every i= 1, . . . , n. Thus,
n \
i=1
Mαi ⊇Mγ 6=∅ by
Claim 1. Claim 3.
Now we can show that G is amenable. Since [0,1]P(G) is compact (by Tychonoff’s theorem) and {Mα}α∈I is a family of closed sets with the finite intersection property,
then \
α∈I
Mα 6=∅. Let µ∈ \
α∈I
Mα, it is clear that µ is a finitely additive measure and
thatµ(G) = 1 (sinceµ∈ Mα), so it remains to show thatµisG-invariant. Letg ∈Gand
A⊆G, then there is an α∈I such that g ∈Gα. Hence, since µ∈ Mα, µ(A) =µ(g·A).
We conclude that Gis amenable.
Lemma 1.3.4. Every group is the direct limit of their finitely generated subgroups.
i ≤ j if and only if i is a subgroup of j, and the homomorphism fij : i → j are the
inclusion maps. Firstly, we will show that (I,{fij}ij) is a directed system:
• Leti, j, k ∈ I such that i≤j ≤ k, then fij(fjk(x)) =fij(x) =x=fik(x) and fii is
the identity of i.
• Leti, j ∈ I and let{g1,· · · , gm}and{g′1,· · · , gl′}be their respective generators. Let
k be the group generated by {g1,· · · , gm, g′1,· · ·, g′l}. Clearly iand j are subgroups
of k.
We can define H = lim
→ i=
G
i/∼, where for xi ∈i ,xj ∈j, we say that (xi, i)∼ (xj, j)
if and only if ∃k ∈ I such that fik(xi) = fjk(xj) (they are related if eventually there are
the same element in the directed system). And let φi :i→ H defined asφi(x) = [(x, i)]∼.
Now consider the inclusion maps ψi : i → G, and notice that ∀i, j ∈ I, φj ◦fij = φi.
Therefore, by the universal property of H there is an homomorphism u : H → G, such that for all i∈ I,u◦φi =ψi, and since ψ is injective u is an embedding. Observe thatu
is also surjective, because for some g ∈G we can consider i=hgi, and since u◦φi =ψi,
then g =φi(g) =u◦φi(g) =u([(g, i)]∼).
Theorem 1.3.5. Every abelian group is amenable.
Proof. We start by showing that every finitely generated abelian group is amenable. Let ǫ >0 andGan abelian group generated by {g1, . . . , gm}. If µ∈[0,1]P(G) we will say that
µis an ǫ-invariant if:
• µ(G) = 1,
• µis a finitely additive measure,
• For all A⊆Gand gi a generator, then |µ(A)−µ(gi·A)| ≤ǫ
Define Mǫ = {µ ∈ [0,1]P(G) | µis an ǫ- invariant measure}. Firstly, we will show that
LetN ∈Nbe such that N2 < ǫand defineµǫ :P(G)→[0,1] asµǫ(A) = |{i∈N |0≤i≤N g
i∈A}|
N .
Notice that µǫ(G) = NN = 1, because gi ∈ G for 1 ≤ i ≤ N. Also, µǫ(∅) = 0. Consider
now A, B disjoint subsets of X. If gi ∈ A for some 0 ≤ i ≤ N then gi ∈/ B. Therefore
µ(A∪B) =µ(A) +µ(B).
Finally, notice that |µ(A)−µ(g·A)| ≤ |{1, N}|
N =
2
N ≤ǫ, since for all 1≤i < N we have thatgi ∈Aif and only if gi+1 ∈g·A(but it might happen thatg ∈g·AorgN ∈A).
It is possible to show that Mǫ is a closed subset of [0,1]P(G) in a similar way as
in the proof of Proposition 1.3.3. Now, consider the family F = {Mǫ}ǫ>0, and
ob-serve that F has the finite intersection property. In fact, for ǫ1,· · · , ǫn we can take
ǫ = min{ǫi | 1 ≤i ≤n} and ∅ 6=Mǫ ⊆ Tni=1Mǫi. Therefore, since [0,1]
P(G) is compact there is an element µ ∈ Tǫ>0Mǫ, which will be a finitely additive measure satisfying
µ(G) = 1 and for every A ⊂ G, µ(A) = µ(g·A). So, for every A⊆ G, and h =gn ∈ G,
we have µ(A) =µ(g·A) =· · ·=µ(g· · ·g
| {z } n−times
A) =µ(h·A). We conclude thatGis invariant. If G has {g1,· · · , gm}generators, let N be such that N2m < ǫ and define
µǫ(A) =
|(i1,· · · , im)|1≤i1,· · · , im ≤N g1i1· · ·gimm ∈A|
Nm .
Note that we needG to be abelian in order to show that µǫ is well-defined (one can write
the finite products of generators as g1i1· · ·gim
m ) and to showǫ-invariance of µǫ.
Now, if G is an abelian group, then every finitely generated subgroup of G is also abelian and thus amenable by the first part of this proof. By Lemma 1.3.4, Gis a direct limit of its finitely generated subgroups, and since they are all amenable, by Proposition 1.3.3, we conclude that G is amenable.
Definition 1.3.6 (Abelian-by-finite). Let (G,·, e) be a group, we say that G is abelian-by-finite if there is a normal abelian subgroup N of G such that [G:N] is finite.
Proof. Let G be a group and suppose that there is a normal subgroup such that N is abelian and its index in G is finite. Then by Theorem 1.3.5, N is amenable and by the Lemma 1.3.2 (1), G/N is amenable. Finally, by Lemma 1.3.2 (4) G is amenable.
Analysis of definable groups in
Presburger Arithmetic
The main goal of this thesis is to understand the definable groups in Presburger Arith-metic. In particular, we are interested in finding a characterization of G/G00 where G is a Presburger-definable group. In order to reach a good understanding of the definable groups we recall a few important results that simplify the description of definable sets, such as quantifier elimination and cell decomposition theorem. The first one is a folklore result which can be found in [12], while the latter one is a theorem proved by R. Cluckers in [3].
Since these results will be very useful to characterize definable sets, we decided to present a detailed exposition of the ideas developed by Cluckers and Marker, for a sake of completeness. We found new proofs for some of the technical lemmas.
In the second part of this section we presented two notions of dimension for the Presburger definable sets, similar to the o-minimal context: one comes from the cell de-composition theorem and the second arises from definable closure. Cluckers introduced these two notions in [3] and he proved they coincide. We presented here a detailed proof
of this fact, which will allow us to extend some useful tools from the o-minimal setting to the Presburger setting, these facts are known as folklore.
Through this chapterT will denote the theory of the structure (Z, <,+,0,1) and M
will be a very large saturated model of T, andG will always denote a definable group in Presburger Arithmetic.
2.1
Presburger Language and quantifier elimination
Proposition 2.1.1. The theory T = Th(Z,+, <,0,1) does not eliminate quantifiers in the language L−={+, <,−,0,1}
Proof. Define the formulas E(x) = ∃z(z +z = x) and O(x) = ∃z(z +z + 1 = x), and consider the two-type Γ(y1, y2) given by
Γ(y1, y2) ={E(y1), O(y2)} ∪ {y1, y2>1 +· · ·+ 1
| {z }
n−times
:n < ω}.
Clearly Γ(y1, y2) is finitely satisfiable, and by compactness, there are elementsb1, b2 ∈ M
satisfying Γ(y1, y2). Notice that tpL−(b1) 6= tpL−(b2) because M |= E(b1) ∧ O(b2) ∧
∀x(E(x)↔ ¬O(x)).
Let M1 and M2 be the L−-substructures of M generated by b1 and b2, respectively.
Notice that Mi is the subgroup of M generated by 1, bi, for i = 1,2. Thus, we can take
the map f :M1 →M2 given by f(m+nb1) = m+nb2 for anym, n∈Z. This will define an isomorphism between M1 and M2, and since elementary maps preserve quantifier-free formulas, we conclude that qftpL−(b1) = qftpL−(b2). This shows that Presburger
arithmetic does not have quantifier elimination in the language {+, <,0,1}.
Remark 2.1.2. In fact, in the proof the previous proposition, it can be shown thatM1 is isomorphic to the structure Z2 =hZ⊕Z,+, <lex,0Z2
is the lexicographical order in Z⊕Z, since we can identify b1 with (1,0).
Consider the languageLP res={+,−, <,{≡n}n∈N,0,1}, where≡n is a binary relation
defined as x≡ny :=∃z(z +· · ·+z
| {z }
n−times
=x−y).
The theory Th(Z,+, <,0,1) has quantifier elimination in the languageLP ress. In fact,
the theory P r for Presburger arithmetic can be axiomatized by the following sentences:
(i) Axioms for ordered abelian groups.
(ii) 0<1.
(iii) ∀x(x ≤0∧x≥1).
(iv)n ∀x(x ≡n y↔ ∃z(x−y=z+· · ·+z
| {z }
n−times
) for n = 2,3, . . .
(v)n ∀x
n−1
_
i=0
x≡n1 +· · ·+ 1
| {z }
i−times
∧^
j6=i
¬x≡n1 +· · ·+ 1
| {z }
j−times
.
We refer the reader to [12] for more details.
2.1.1
Some consequences of quantifier elimination
The following statements are immediate consequences from quantifier elimination, and will be very important for understanding the decomposition of definable sets into cells presented by Cluckers in [3]. We believe that these results are folklore, but nevertheless we proof then ourselves.
2.1.2
A
-definable functions and
dcl
(
A
)
Let A be a small set of parameters of M. In this section we want to use quantifier elimination to find and explicit characterization of the complete 1-types overA. Moreover, we use this theorem to describe an elementb ∈dcl(A) as the image of anA- linear function.
Definition 2.1.3 (A-linear function). Let f : X ⊆ Mm → M be a function. We say
that f isA-linear, if it can be written in the form
f(x) = m X i=1 si
xi−ci
ki
+γ,
where γ ∈ dcl(A), and for each 1 ≤ i ≤ m, 0 ≤ ci < ki and s are integers , such that
xi ≡ki ci.
Fact 2.1.4. Let x be a single variable and a= (a1, . . . , ak) be a tuple of elements in M.
Then, the terms τ(x, a) are of the form:
sx+
n X
i=1
kiai+n
where si, ki, n are integers.
Proof. We proceed by induction on the length of τ(x, a).Note that the terms depending on a of length 1 are 0,1, x, a1, . . . , ak, which are of the desired form.
Induction Hypothesis: Let σ(x, a) be a term of lenght less than n. Then, σ(x, a) is a Z-linear combination of x and 1, a1, . . . , ak.
Let now τ(x, a) be a term of length n. Assume thatτ(x, a) =σ1(x, a) +σ2(x, a). By the induction hypothesis:
τ(x, a) = s1x+
n X
i=1
ki1ai+n1 !
+ s2x+
n X
i=1
ki2ai+n2 !
Then,
τ(x, a) = s1+s2
| {z }
∈Z
x+
n X
i=1
k1i +ki2
| {z }
∈Z
ai+ n1+n2
| {z }
∈Z