Group construction in geometric C-minimal non-trivial structures.

Transcription

1 Group construction in geometric C-minimal non-trivial structures. Françoise Delon, Fares Maalouf January 14, 2013 Abstract We show for some geometric C-minimal structures that they define infinite C-minimal groups. 1 Introduction In this paper we show the following. Theorem: Let M be a C-minimal non-trivial geometric definably maximal structure. Suppose moreover that in the underlying tree of M there is no definable bijection between a bounded interval and an unbounded interval. Then M defines an infinite C-minimal group. This result is established as a part of the Zilber s trichotomy problem for geometric C-minimal structures, without assuming any kind of linearity on the geometry of the structure. The linear was solved in [4] and [5], and a particular case with non-linearity assumptions was treated in [6]. We start with a few definitions and preliminary results. C-structures have been introduced and studied in [3] and [7]. We remind in what follows their definition and principal properties. Notations: We use M, N,... to denote structures and M, N,... for their underlying sets. A C-structure is a structure M = (M, C,...), where C is a ternary predicate satisfying the following axioms: x, y, z, C(x, y, z) C(x, z, y) x, y, z, C(x, y, z) C(y, x, z) x, y, z, w, C(x, y, z) C(x, w, z) C(w, y, z) x, y, x y z y, C(x, y, z) Let M be a C-structure. We call cone any subset of M of the form {x; M = C(a, x, b)}, where a and b are two distinct elements of M. It follows from the first three axioms of C-relations that the cones of M form a basis of a completely disconnected topology on M. The last axiom guarantees that all cones are infinite. Let (T, ) be a partially ordered set. We say that (T, ) is a tree if the set of elements of T below any fixed element is totally ordered by, and if any two elements of T have a greatest lower bound. A branch of T is a maximal totally ordered subset of T. It is easy to check that if a and b are two distinct branches of T, then sup(a b) exists. On the set of branches of T, we define a ternary relation C in the following way: we say that C(a, b, c) is true if and only if b = c 1

2 or a, b, and c are all distinct and sup(a b) < sup(b c). It is easy to check that this relation on the set of branches satisfies the first three axioms of a C-relation. A theorem from [1] shows that C-structures can be looked at as a set of branches of a tree. We will then associate to any C-structure M a tree T. We will call it the underlying tree of M, and the elements of T will be called nodes. To any x, y M, we associate the node t := sup(x y), where x and y are seen as subsets of T. This operation is well defined, and we say then that x and y branch at t. If a and b are two elements of M branching at a node t, and if D is the cone D := {x M; C(a, x, b)}, we say then that D is the cone at the node t containing b. Definition 1.1. Let M = (M, C,...) be a C-structure. 1. The structure M is said to be geometric if the algebraic closure in M has the exchange property. 2. The structure M is said to be definably maximal if any uniformly definable family of cones which is completely ordered by the inclusion has a non-empty intersection. 3. The structure M is said to be C-minimal if and only if for any structure M = (M, C,...) elementarily equivalent to M, any definable subset of M can be defined without quantifiers using only the relations C and =. Setting: We fix a definably maximal C-minimal geometric structure M with universe M. We suppose that in the underlying tree of M there is no definable bijection between a bounded interval and an unbounded interval. (bounded from the two sides). In Section 2, we show that M defines a family of functions with nice properties. The idea is to get a group law by composing the elements of this family. For this we need a tool, the derivatives, which we define and study in Sections 3 and 4, and which we use in Section 5 to construct an infinite group in M. 2 Nice families of functions. Lemma 2.1. Let V be a cone, e be an element of V, and f, g, h : V M be definable functions. Then there is a neighbourhood W of e such that either any element x W \ {e} satisfies C(f(x), g(x), h(x)), or any element x W \ {e} satisfies C(f(x), g(x), h(x)). Proof. Let D M be the set of elements x of V such that C(f(x), g(x), h(x)) holds. So either e is a limit point of D, in which case by C-minimality there is a neighbourhood W of e such that x W \ {e} C(f(x), g(x), h(x)), or e is in the interior of the complement of D, so there is a neighbourhood W of e such that x W \ {e} C(f(x), g(x), h(x)). Notation: 1. Let x, y, z M. We denote by (x, y, z) the property C(x, y, z) C(y, x, z). 2. Let V M be a cone, e be an element of V, and f, g, h be functions from V to M. We denote by C e (f, g, h) (or C(f, g, h) if there is no confusion on e) the following property: there exists a neighbourhood W of e such that x W \ {e} : C(f(x), g(x), h(x)). 2

3 We define ( C) e (f, g, h) and e (f, g, h) in the same way, and we denote them by ( C)(f, g, h) and (f, g, h) respectively, if there is no confusion on e. Definition 2.2. Let V be a cone in M, e V and f : V M be a definable function. 1. The function f is said to be dilating on a neighbourhood of e, or just dilating if there is no confusion, if it satisfies C e (f 2, f, id V ). 2. The function f is said to be contracting on a neighbourhood of e, or just contracting if there is no confusion, if it satisfies C(f 2, f, id V ). Definition 2.3. Let F = {f u : u U} be a uniformly definable family of definable functions, indexed by a cone U M. The family F is said to be a nice family of functions if there is a cone V M, and an element e V with the following properties: 1. All the f u are C-automorphisms of the cone V. 2. For every u U, we have f u (e) = e. 3. For any fixed x V \ {e}, the application U V which to u associates f u (x), is a C-isomorphism onto some subcone of V. ( ) A nice family F of functions is said to have an identity element if for some u 0 U, f u0 = id V. Notation: If F is a nice family of functions, possibly with identity, then U, V, e and u 0 will be as in Definition 2.3. Terminology: For a nice family of functions F, the sets U and V are called respectively the index set and the domain, and e is called the absorbing element. The set V \ {e} will be denoted by V. Proposition 2.4. Let M be a geometric non trivial C-minimal structure. Then M defines a nice family of contracting functions with identity. Proof. By Proposition 15 and Lemma 21 of [4], we can find cones U 1, V in M and a uniformly definable family of functions H = {h u : u U 1 } of C-automorphisms of V satisfying ( ). Fix a, b U, e V, and let e := h a h b (e). For every u U 1, let θ(u) be the unique z U 1 (when it exists) such that h z h u (e) = e. The function θ is well defined on some cone U 2 containing b. For u U 2, define g u := h θ(u) h u. Now let u 0 be any element of U 2, and define f u := gu 1 0 g u for any u U 2. The family F := {f u : u U 2 } is a family of C-automorphisms of V having the property ( ), f u (e) = e for every u U 2, and f u0 = id V. So F is a nice family of functions with identity. By C-minimality, there is a neighbourhood U of u 0 such that, either for all u U, the function g u is contracting, or for all u U, the function g u is dilating. We can always suppose that we are in the former case, by replacing all the g u by gu 1 if necessary. 3 Tangents: existence and uniqueness Definition 3.1. Let F be a nice family of functions, and g be a C-automorphism of V such that g(e) = e. 1. Let u be an element of U. The function g is said to be tangent to f u relatively to the family F if for any u U, we have ( C)(f u, f u, g), in which case we write f u F g, or just f u f v if there is no confusion on F. 3

4 2. The g is said to be derivable relatively to the family F if there is a unique f u such that f u g. In this case, f u is called the derivative of g relatively to the family F. We fix a nice family F of functions. Let g : V V be a fixed definable function such that g(e) = e. We define and for u U, we define T g := {u U; f u g}, Γ u := {y U : C(f u, f y, g)}. For a node ν of the underlying tree of M, we denote by Λ(ν) the closed ball of M defined by ν. This corresponds to the set of all branches of the tree containing ν. Lemma Let u and v be two distinct elements of T g. Then Λ(u v) T g. Furthermore, there is a cone W containing e such that z Λ(u v) x W : (f u (x), f z (x), g(x)). 2. Let u T g and v U \T g. Then the cone Γ of v in u v is contained in U \T g. Furthermore, there is a cone W containing e such that 3. T g is a ball. z Γ, x W : C(f z (x), f u (x), g(x)). Proof. 1. Let u, v be two distinct elements of T g, and let W be a neighbourhood of e such that, x W : (f u (x), f v (x), g(x)). Let w Λ(u v), so we have C(w, u, v). By ( ), we have that x W : C(f w (x), f u (x), f v (x)). So C(f w (x), f u (x), g(x)) holds for all x W, and it follows that w T g. assertion, let z be an element of Λ(u v) T g. Let For the second W z := {W : W V is a cone, e W, x W : (f u (x), f z (x), g(x))}. W z is a non empty union of cones, so by C-minimality it is a ball. Let ν z be the basis of W z. If z U is such that C(u, z, z), then (z T g and) ν z = ν z. Thus the application z ν z induces an application from the set of cones at u v to the branch of e. By orthogonality between strong minimality and linear ordering, the image of this application is finite. Let ν be the maximal element of the image. So we have 2. Same proof. z Λ(u v), x Λ(ν) : (f u (x), f z (x), g(x)). 3. By 1, T g is a union of closed balls. It is then a cone or a closed ball by C-minimality. Lemma 3.3. Γ u = if and only if u T g. Proof. Fix the elements u and v. So either ( C)(f u, f v, g), or C(f u, f v, g). By the definition of tangent, u T g if and only if we are in the first case for every v U, if and only if Γ u =. Lemma 3.4. Let u, u, v, v be elements of U such that v Γ u and v Γ u. Then if v is not an element of Γ u, then v is an element of Γ u. 4

5 Proof. Let u, v, u, v be elements of U such that v Γ u, v Γ u and v / Γ u. Let W V be a neighbourhood of e such that, for every x W, we have C(f u (x), f v (x), g(x)), C(f u (x), f v (x), g(x)) and C(f u (x), f v (x), g(x)). By the first and third relations, we have This together with the second relation yields So v is an element of Γ u. x W : C(f u (x), f u (x), g(x)). x W : C(f u (x), f v (x), g(x)). Lemma 3.5. If Γ u is not empty, then it is a cone on a node on the branch of u. The non empty Γ u form a chain of cones. Proof. Let v Γ u. By an argument similar to that in the proof of Lemma 3.2, it is easy to see that if w is an element such that C(u, v, w), then w Γ u. This shows that the cone of v at u v is contained in Γ u. On the other hand, by a similar argument one shows that if x and y are elements of Γ u, then Λ(x y) Γ u. So by C-minimality, Γ u is a ball, and since it contains the cone of v at u v, then it is a ball at a node ν on the branch of u. But it is clear that u / Γ u. Thus ν = u v and Γ u is the cone of v at the node u v. The claim that the non empty Γ u form a chain of cones follows directly from Lemma 3.4 and the fact that in C-structures, two cones have a nonempty intersection if and only if one is a subcone of the other. Lemma 3.6. There is an element u U such that f u g. Proof. If T g is empty, then by Lemma 3.3, no cone Γ u is empty. By Lemma 3.5, the Γ u form a chain of cones of U. By maximal definability, their intersection is not empty. But this intersection is contained in T g, which is a contradiction. Lemma For every u T g there is a cone W containing e such that (f u (x), f z (x), g(x)) holds for all z T g and all x W \ {e}. 2. Suppose that T g contains more than one element, and let u T g. Then there is a cone W containing e such that C(f z (x), f u (x), g(x)) holds for all z U \ T g and all x W \ {e}. Proof. 1. The claim is trivial if T g = {u}, since (x, x, y) holds for all x and y. If T g is not a singleton, then c 0 := inf T g is a well defined node of the underlying tree of M, since T g is a subcone of the cone U. For u T g, and any node c c 0 of the form u v for some v T g, let W c,u := {W : W V is a cone, e W, x W \ {e}, z Λ(c), (f u (x), f z (x), g(x))}. By Lemma 3.2 (1), W c,u is not empty, and by C-minimality, W c,u is a ball. Let ν c,u be the BASIS of W c,u. If z Λ(c), then (z T g and) ν c,u = ν c,z =: ν c. REVIRIFIER So we have z, z Λ(c), x Λ(ν c ) (x e (f z (x), f z (x), g(x))). If T g is a closed ball, we are done by taking for W the cone of e at the node ν c0. If not, fix a node c 1 strictly above c 0. The application which to a node c in the interval (c 0, c 1 ] associates ν c is a decreasing application from a bounded interval of the tree to the chain of e. So by o-minimality and the fact that in the tree, there are no definable cofinal applications between bounded intervals and unbounded intervals, the image of the above application is bounded from above by a certain node d. For every x in the cone of e at d, we have z T g, (f u (x), f z (x), g(x)). 5

6 2. Suppose that T g is a ball containing at least two elements, and let c 0 := inf T g. Fix an element a T g, and a cone W containing e such that (f u (x), f z (x), g(x)) holds for all z T g and all x W \ {e}. For any node c c 0, and any element z / T g such that z a = c, there is a maximal ball W c,z W such that x W c,z : C(f z (x), f a (x), g(x)). It is also easy to check that x W c,z : C(f z (x), f a (x), g(x)) for every z such that C(a, z, z ). Since W c,z W, we have u, v T g, x W z : (f v (x), f u (x), g(x)) C(f z (x), f u (x), g(x)) for every z such that C(a, z, z ). Let ν c,z be the basis of W c,z. So the application z ν c,z induces an application from the set of cones at c not containing T g to the branch of e. By orthogonality between strong minimality and linear ordering, the image of this application is finite. Let ν c be the maximal element of the image. Now the application which to c associates ν c is an application from a bounded interval of the tree (namely the interval delimited by the basis of U and that of T g ) to the chain of e. Its image is then bounded from above by some node d. If W 0 W V is the cone of e at the node d, then we have u, v T g, z U \ T g, x W 0 : (f v (x), f u (x), g(x)) C(f z (x), f u (x), g(x)). Proposition 3.8. The following are equivalent: 1. there are elements u, v U such that C(f u, f v, g); 2. T g U; 3. g is derivable relatively to the family F. Proof. We show that 3 follows from 1, the rest is trivial. Let u, v be like in the statement of 1. We know that T g is not empty. Suppose that it contains two distinct elements. By Lemma 3.2, T g is a cone or a closed ball. By Lemma 3.7, there is a cone W containing e such that α T g, z U, x W : C(f α (x), f z (x), g(x)). (a) Restricting W if necessary, we can suppose that x W : C(f u (x), f v (x), g(x)). Fix an element x 0 W. By ( ), there is an element w U such that f w (x 0 ) = g(x 0 ). By (a), w T g, and for every α T g, f α (x 0 ) = g(x 0 ). But T g contains at least two elements, this contradicts ( ). 4 Derivability relatively to nice families of functions We fix a nice family F of functions. Lemma 4.1. Let g be a C-automorphism of V such that g(e) = e. Suppose that g is derivable relatively to the family F, and let f u1 be its derivative. Then for every u U, u u 1, we have C(f u, f u1, g). Proof. 6

7 For x V \ {e}, the function φ F,x := φ F,x (U). { U V u f u (x) is a C-isomorphism from U onto a cone Let g be a C-automorphism of V such that g(e) = e. We define the function { V \ {e} U ψ F,g := x φ 1 F,x (g(x)) So ψ F,g (x) is the unique element u of U such that f u (x) = g(x), when such an element exists. Lemma 4.2. If g is derivable relatively to the family F, then ψ F,g is defined on a neighbourhood of e. Proof. Let f u1 be the derivative of g, and let u u 1 be an element of U. By Lemma 4.1, we have C(f u, f u1, g). So for some neighbourhood W of e, we have x W : C(f u (x), f u1 (x), g(x)). For every x W, f u (x) and f u1 (x) are elements of the cone φ F,x (U), so the same holds for g(x). Hence ψ F,g is defined on W. Proposition 4.3. If g is derivable relatively to F, then f u1 g if and only if lim ψ F,g(x) = u 1. x e Proof. Let u 1 U be such that f u1 g. If f u1 = g on a neighbourhood of e, then ψ F,g is defined and takes the constant value u 1 on this neighbourhood. Suppose then that on any neighbourhood of e, f u1 and g define distinct functions. By C-minimality, there is a neighbourhood W 0 of e such that x W 0 \ {e} : g(x) f u1 (x). By Lemma 4.2 we can, by reducing W 0 if necessary, suppose that ψ F,g is defined everywhere on W 0 \ {e}. Fix an element u U \ {u 1 } and let W 1 be a neighbourhood of e such that x W 1 \ {e} : C(f u (x), f u1 (x), g(x)). (1) For every x W 0 W 1 \{e}, we have that f ψf,g (x)(x) = g(x), and g(x) f u1 (x). So ψ F,g (x) u 1. Furthermore, by (1) and the properties of F follows that C(u, u 1, ψ F,g (x)). We showed that for any u u 1, there is a neighbourhood W := W 0 W 1 of e such that x W : C(u, u 1, ψ F,g (x)). This shows that the unique possible limit of ψ F,g at e is u 1. On the other hand, we know by maximal definability and Proposition 4.4 of [2] that ψ F,g has a limit at e. So we have that lim ψ F,g(x) = u 1. x e Definition 4.4. Let F and G be two nice families of functions having the same domain and absorbing element. 1. The family G is said to be derivable relatively to F if every element of G is derivable relatively to F. 7

8 2. The families F and G are said to be comparable if both are derivable relatively to each other. Notation: Let F : {f u : u U} and G = {g u : u U } be two families of functions. Suppose that the family G is derivable relatively to the family F. We define the derivative function and denote it by F,G the function { U U F,G := u u such that f u g u. Lemma 4.5. Let F : {f u : u U} and G = {g u : u U } be two nice families of functions. Suppose that the family G is derivable relatively to the family F. Let a 1, a 2 and a 3 be elements of U such that C(a 1, a 2, a 3), and let a i := F,G (a i ). Then exactly one of the following holds: C(a 1, a 2, a 3 ), or a 1 = a 2 = a 3. Proof. It is enough to show that it is impossible to have C(a 1, a 2, a 3 ) Suppose for a contradiction that C(a 1, a 2, a 3 ) holds. Let W be a neighbourhood of e such that, for every x W we have and Therefore, for every x W holds: C(f a3 (x), g a 1 (x), f a1 (x)) C(f a3 (x), g a 2 (x), f a2 (x)) C(f a1 (x), g a 3 (x), f a3 (x)) C(f a2 (x), g a 3 (x), f a3 (x)). C(g a 1 (x), g a 2 (x), g a 3 (x)). By ( ) we have: contradiction. C(a 1, a 2, a 3), Corollary 4.6. Let F : {f u : u U} and G = {g u : u U } be as above, and let a be an element of U. Then F,G (x) admits a limit b U, when x converges to a. Furthermore, lim x a F,G (x) = F,G (a ), and F,G is continuous on U. Proof. By maximal definability, Proposition 4.4 of [2] and Lemma 4.5, the function F,G admits a limit in each point of U, and the limit is an element of U. For the second assertion, let a U and let a := F,G (a ). If the limit of F,G in a is an element b a, then we can find elements b 1, b 2 U such that C(b 1, b 2, a ), and C(a, b i, b) for i = 1, 2, where b i = F,G (b i ). This contradicts Lemma 4.5. Proposition 4.7. Let F : {f u : u U} and G = {g u : u U } be two nice families of functions. Suppose moreover that G is derivable relatively to F. Let a, a be elements of U and U respectively. Then f a g a if and only if for any neighbourhood A U of a, any neighbourhood A U of a, and any neighbourhood E V of e, there are elements α A, α A and y E such that f α (y) = g α (y). Proof. Fix elements a U and a U and suppose that f a g a. For the first direction, let A, A and E be as in the statement of the theorem. By Lemma 4.3, we have that lim x e ψ F,g a (x) = a. 8

9 Since A and E are neighbourhoods of a and e respectively, we can find an element y E such that ψ F,ga (y) A. Set α := ψ F,ga (y) and α := a. We have then that α A, α A, y E, and by the definition of ψ we have f α (y) = g α (y). For the converse, let b U, b a, and let E 0 be a neighbourhood of e such that x E 0 : C(f b (x), g a (x), f a (x)). (1) Let Γ be the cone of a at the node a b. The element F,G (a ) = a is an element of Γ, so by continuity of F,G, there is an element a 1 U \ a such that a 1 := F,G (a 1) Γ. Let E 1 V be a neighbourhood of e such that x E 1 : C(f b (x), g a 1 (x), f a1 (x)). (2) Let A be the cone of a at the node a a 1, A be the cone of a at the node a a 1, B be the cone of b at the node a b, and E := E 0 E 1. By the properties of F, we have β B, x E : C(f a (x), f β (x), f b (x)). (3) By (1), (2) and the fact that C(f b (x), f a1 (x), f a (x)) for any x, we have x E : C(f b (x), g a (x), f a (x)) C(f b (x), g a 1 (x), f a (x)). (4) Furthermore, by the properties of G we have: From (4) and (5) we have By (3) and (6), we have α A, x E : C(g a 1 (x), g α (x), g a (x)). (5) α A, x E : C(f b (x), g α (x), f a (x)). (6) β B, α A, x E : f β (x) g α (x), and B, A and E are neighbourhoods of b, a and e respectively. This completes the proof. An immediate consequence of Proposition 4.7 is the Corollary 4.8. Let F = {f u : u U} and G = {g u : u U } be two comparable families of functions, and let a, a be elements of U and U respectively. Then f a g a if and only if g a f a. Corollary 4.9. Let F : {f u : u U} and G = {g u : u U } be two comparable families of functions. Then F,G and G,F are inverse of each other, and they define (continuous) C- isomorphisms between U and U. Proof. By Corollary 4.8, the functions F,G and G,F are inverse of each other, thus define bijections between U and U. Continuity follows by Corollary 4.6, and the rest by Lemma 4.5. Example. Let M be an Algebraically closed valued field with maximal ideal Υ, U = U := 1+Υ, V := Υ, u 0 = u 0 = 1, e = 0, F := {u.x; u U} and G := {x + (u 1).x 2 ; u U }. Then G is derivable relatively to F, and F,G is constant and sends every u to 1. Now G,F is defined only in one point 1, and its image in this point is 1. Proposition Let F : {f u : u U}, G := {g u : u U } and H := {h u : u U } be comparable families of functions. Then H is derivable relatively to F, and we have: F,H = F,G G,H. 9

10 Proof. Let a, b, c be elements of U, U and U respectively, such that f a g b and g b h c. It is enough to show that f a h c. Suppose first that C(h c, f a, g b ) holds. If f a1 h c for some a 1 a, then we have (f a1, f a, g b ). So both f a1 and f a are derivatives of g b, and this contradicts the uniqueness of the derivative. Now suppose that C(h c, f a, g b ), and that f a1 h c for some a 1 a. For every y b, we have C(g y, g b, h c ), thus C(g y, f a1, g b ), and g b f a1. By Corollary 4.8 follows f a1 g b, so a 1 = a. Contradiction. 5 The group law Lemma 5.1. Let F := {f u : u U} be a nice family of contracting functions with identity, and let a, b U. Let g, h be two C-isomorphisms of V with g(e) = h(e) = e, and assume that there is an element c U such that C(f c, g, f u0 ) and C(f c, h, f u0 ). Then we have: 1. C(f c, g h, f u0 ), and g h is derivable relatively to the family F. 2. C(g f c, h, g), and h is derivable relatively to the family {g f u : u U} (when this is a nice family of functions). 3. C(f c g, h, g), and h is derivable relatively to the family {f u g : u U} (when this is a nice family of functions). Proof. 1. Let E 1 be a neighbourhood of e such that x E 1 : C(f c (x), g(x), x) C(f c (x), h(x), x) C(f 2 (x), f(x), x). Let E E 1 be a neighbourhood of e such that f c (E) E 1. The relations and C(g f c (x), g h(x), g(x)) C(f c f c (x), g f c (x), f c (x)) hold for all x E. From the second and C(f 2 (x), f(x), x) follows C(x, g f c (x), f c (x)). Now it is clear that C(f c (x), g h(x), x) holds for every x in E, and we have C(f c, g h, f u0 ). The derivability of g h relatively to the family F follows by Proposition We have C(f c f c, g f c, f c ) (a), C(f c, h, g) (b) and C(f 2, f, id) (c). By (a) and (c) follows C(id, g f c, f c ), which, together with C(f c, h, id) yields C(h, g f c, f c ). From this last relation and (b) follows C(g f c, h, g). Derivability follows as above by Proposition Like above. Use that C(f c f c, f c g, f c ), C(f c, h, g) and C(f 2, f, id), one sees easily that C(f c g, h, g) holds. We fix a nice family of contracting functions F := {f u : u U } with identity, where U is an open ball. Let c u 0 be an element of U, and U := {x U : C(c, x, u 0 )}. Then the family F := {f u : u U} is a nice family of contracting functions with identity. For every a U, the families {f a f u : u U}, {fa 1 f u : u U}, {f u f a : u U} and {f u fa 1 : u U} are nice family of functions. Furthermore, Lemma 5.1 yields that they are comparable with F: for any a, b U, we have C(f c, f a, f u0 ) and C(f c, f b, f u0 ). Notations: If g is derivable relatively to F, we denote by ĝ its derivative. We define an operation ô from F F F by f u ôf v := f u f v. By the above argument, ô is well defined. We will show now that (F, ô, f u0 ) is a group (which is clearly infinite). 10

11 Lemma 5.2. The operation ô is regular. Proof. We show right regularity, left regularity can be done in a similar way. Suppose that f w = f u ôf a = f v ôf a. So f w f u f a, and f w f v f a. Let G := {f x fa 1 : x U}. So f w fa 1 G f u. By Corollary 4.8 we have f u f w fa 1. The same argument yields f v f w fa 1, and we have f u = f v. Remark 5.3. If g is a such that f a g, then f a f v G g f v, where G := {f a f x : x U}. CHECK Furthermore, by the definition of ô, we have f a ôf v f a f v. So by Proposition 4.10 we have f a ôf v g f v. The same is true when we compose with g from the right: f v ôf a f v g. This means with our notations: ĝ ôf v = ĝ f v and Lemma 5.4. The operation ô is associative. f v ô ĝ = f v g. Proof. By the above remark we have the following: (f a ôf b )ôf c = (f a f b )ôf c = (fa f b ) f c = f a f b f c and So ô is associative. f a ô(f b ôf c ) = f a ô( f b f c ) = fa (f b f c ) = f a f b f c. Lemma 5.5. Every f a F admits an inverse. Proof. Let G be the family of all the f a f u : u U. The families F and G are comparable, so let b U be such that f a f b G f u0. By Corollary 4.8 we have f u0 f a f b, so f a ôf b = f u0, which is the identity element of (F, ô). So f a has a right inverse, and the same argument shows that f a has a left inverse. We have proved the following Proposition 5.6. Let M be a C-minimal geometric definably maximal structure. Suppose moreover that in the underlying tree of M there is no definable bijection between a bounded interval and an unbounded interval, and that M defines a nice family of contracting functions with identity. Then M defines an infinite C-minimal group. Propositions 2.4 and 5.6 yield directly the following Theorem. Theorem 5.7. Let M be a C-minimal non-trivial geometric definably maximal structure. Suppose moreover that in the underlying tree of M there is no definable bijection between a bounded interval and an unbounded interval. Then M defines an infinite C-minimal group. References [1] S.A. Adeleke and P.M. Neumann, Relations Related to Betweennes: Their Structure and Automorphisms, American Mathematical Society (1998). [2] F. Delon, Corps C-minimaux en l honneur de François Lucas, Annales de la Faculté des Sciences de Toulouse XXI (2012), [3] D. Haskell and D. Macpherson, Cell Decompositions of C-minimal Structures, Ann. Pure Appl. Logic 66 (1994), [4] F. Maalouf, Construction d un groupe dans les structures C-minimales J. Symbolic Logic 73 (2008),

12 [5] F. Maalouf, Type-definable groups in C-minimal structures Comptes Rendus de l Académie des Sciences - Mathématiques, 348 (2010): [6] F. Maalouf, Additive reducts of algebraically closed valued fields Forum Mathematicum, (to appear). [7] D. Macpherson and C. Steinhorn, On variants of o-minimality Ann. Pure Appl. Logic 79 (1996),