G-linear sets and torsion points in definably compact groups

Margarita Otero^{∗} Universidad Autónoma de Madrid Ya’acov Peterzil University of Haifa

August 3, 2007

Abstract

Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G \ X) < dim G for some definable X ⊆ G then X contains a torsion point of G. Along the way we develop a general theory for so-called G-linear sets, and investigate definable sets which contain abstract subgroups of G.

Keywords: o-minimality, definable group, torsion point.

Mathematics Subject Classification 2000: 03C64

1 Introduction

We prove:

Theorem 1.1. Let G be a definably compact group in an o-minimal expansion of a real closed field. If X ⊆ G is a definable large set (i.e. dim(G \ X) < dim G) then X contains a torsion point of G.

This is a weak approximation to the conjecture that every generic set in a definably compact group contains a torsion point, a conjecture which itself follows from the “compact domination conjecture” from [4].

The proof starts with the abelian case, where we first develop, in ominimal expansions of an ordered group, a notion of a coset of a definable local subgroup (a G-linear set). We then use tools from Algebric Topology, in expansions of real closed fields, to conclude the theorem.

∗Partially supported by GEOR MTM2005-02568 and Grupos UCM 910444

2 G-linear sets

In this section M will denote an o-minimal expansion of an ordered group. By definable we mean definable in M. G will denote a definable abelian group. All topological concepts are with respect to the group topology such G is equipped with. (We believe that most, if not all, of the work below can be developed in an arbitrary definable group, but for simplicity we limit ourselves to the commutative setting.)

Definition 2.1. 1) Given two sets X, Y ⊆ G and a ∈ G, we say that X and Y have the same germ at a, in notation X =_{a} Y, if there exists an open neighborhood U of a such that X ∩ U = Y ∩ U. We say that the germ of X at a is contained in the germ of Y at a, in notation X ⊆_{a} Y, if there exists an open U ∋ a such that X ∩ U ⊆ Y ∩ U.

2) Given X ⊆ G and g, h ∈ G, we say that the germ of X at g is Gequivalent to the germ of X at h if X − g =_{0} X − h. (Note that if X is a definable set then we obtain in this way a definable equivalence relation on G.)

Definition 2.2. Let X be a definable subset of G.

1) X is called G-linear if for every g, h ∈ X we have X − g =_{0} X − h. Given g ∈ X, we say that X is locally G-linear at g if there exists an open U ∋ g such that for every h ∈ U ∩ X we have X − g =_{0} X − h.

Two G-linear sets X, Y ⊆ G are called G-equivalent if for every g ∈ X, h ∈ Y, we have X − g =_{0} Y − h.

2) Let X be a G-linear set. A definable Y ⊆ G is called a G-subset of X if for every h ∈ Y and g ∈ X, we have Y − h ⊆_{0} X − g.

Example 2.3. Let G be a definable abelian group and H a definable subgroup. If X is a relatively open subset of a coset of H then X is G-linear. If Y is any definable subset of a coset of H then Y is a G-linear subset of X.

We observe without a proof (we will not be using it): If M expands a real closed field then Y is a G-subset of the G-linear set X if and only if at every smooth point y ∈ Y, the tangent space of Y at y is a subset of the tangent space of X at any of its points point.

Lemma 2.4. If X is definably connected and locally G-linear at every g ∈ X then X is G-linear.

Proof. Because X is locally G-linear at every point the G-equivalence class of every g ∈ X is relatively open in X. But then it is also relatively closed (since its complement is a union of open sets), so by definable connectedness there exists only one G-equivalence class, so X is G-linear.

Definition 2.5. Given X ⊆ G, we let

X_{lin} = {g ∈ X : X is locally G-linear at g}.

Notice that X_{lin} is a relatively open (possibly empty) subset of X. By the last lemma, every definably connected component of X_{lin} is a G-linear set, but X_{lin} itself might not be G-linear (E.g., if X = H_{1} ∪ H_{2} is the union of two definable subgroups of G, none containing the other, then X_{lin} = X \ (H_{1} ∩ H_{2}) is not a G-linear set). It is easy to see that X_{lin} is itself G-linear if and only if all its definably connected components are G-equivalent. Notation We denote by X_{lin}^{max} the union of all components of X_{lin} of maximal dimension.

Infinitesimals

It might be easier here to use the language of infinitesimals: We move to an |M|^{+}-saturated elementary extension N of M. For g ∈ G(M), we denote by ν_{g} the intersection of all M-definable open neighborhoods of g (with respect to the group topology) in the structure N. For X ⊆ G, we write ν_{g}(X) = ν_{g}∩X. Notice that for X, Y ⊆ G and g, h ∈ G, ν_{g}(X) = ν_{g}(Y) iff X =_{g} Y, and ν_{g}(X) − g = ν_{h}(X) − h iff X − g =_{0} X − h.

We will be using the following simple observation: If X, Y are A-definable sets, and x is generic in X and in Y over A then ν_{x}(X) ⊆ Y.

Lemma 2.6. X is locally G-linear at g ∈ X if and only if ν_{g}(X) − g is a subgroup of G. In particular, X is a C^{0}-manifold near g.

Proof. : If ν_{g}(X) − g is a subgroup then for every h ∈ ν_{g}(X), we have h − g + ν_{g}(X) = ν_{g}(X). It easily follows that X is locally G-linear at g.

Conversely, assume that X is locally G-linear at g. We need to prove: For all h_{1}, h_{2} ∈ X sufficiently close to g we have (h_{1} − g) − (h_{2} − g) = h′ − g, for some h′ ∈ X, or equivalently, g + (h_{1} − h_{2}) ∈ X.

We pick h ∈ X close to g and generic over g such that X − g =_{0} X − h. We take U ∋ 0, definable over parameters independent of g, h, such that (X − g) ∩ U = (X − h) ∩ U. Because of the genericity of h, there exists a neighborhood V ∋ h, which we may assume is contained in h + U, such that for all h′ ∈ X ∩ V,

(X − h) ∩ U = (X − g) ∩ U = (X − h′) ∩ U.

By adding h to both sides of the last equation, we obtain: For all h′ ∈ X ∩V, X ∩ (h + U) = (X − h′ + h) ∩ (h + U), hence (since V ⊆ (h + U)) also for all h′ ∈ X ∩ V,

X ∩ V = (X − h′ + h) ∩ V.

In particular, if h′′, h′ ∈ X are sufficiently close to h (so that h′′−h′+h ∈ V) then we have (h′′ − h′) + h in X. It follows, as was pointed out above, that ν_{h}(X) − h is a subgroup of G. Because ν_{g}(X) − g = ν_{h}(X) − h, it also follows that ν_{g}(X) − g is a subgroup of G.

Because the germ of X at g is G-equivalent to the germ of X at some generic point near g, it follows that X is a C^{0}-manifold near g.

Notation. For a G-linear X, we denote by ν(X) the group ν_{x}(X) − x, for some (all) x ∈ X. Notice that Y is a G-subset of X iff ν_{h}(Y) − h is a subset of the group ν(X), for every h ∈ Y. Note also that two G-linear sets X_{1}, X_{2} are G-equivalent if and only if ν(X_{1}) = ν(X_{2}).

Lemma 2.7. Let X, Y ⊆ G be definable sets of dimension d. If dim(X + Y) = d then X_{lin} is large in X, Y_{lin} large in Y and X_{lin}^{max}, Y_{lin}^{max} are G-linear sets that are G-equivalent.

Proof. We assume that X and Y are ∅-definable and write Z = X + Y. Take g generic in X over ∅ and h generic in Y over g. Then g + h is generic in Z over each g and h and hence for every two x, y ∈ {g, h, g + h} we have dim(x/y) = d.

For every ∅-definable neighborhood U of g, we have (U ∩ X) + h ⊆ Z. Moreover g + h is generic in (U ∩ X) + h over h and therefore (U ∩ X) + h contains ν_{g+h}(Z). It follows that ν_{g}(X) + h ⊇ ν_{g+h}(Z). By symmetry we may conclude that ν_{g}(X) + h = ν_{g+h}(Z), or equivalently, ν_{g}(X) − g = ν_{g+h}(Z) − (g + h).

In the very same way we see that ν_{h}(Y) − h = ν_{g+h}(Z) − (g + h). Hence, we showed that for every generic g ∈ X, h ∈ Y which are independent from each other we have ν_{g}(X) − g = ν_{h}(Y) − h.

Fixing a generic h ∈ Y, this implies that ν_{g′}(X) is constant as g′ varies over all elements g′ which are generic in X over h. In particular, X_{lin} contains all those elements g′ and moreover all the components of X_{lin} of dimension d are G-equivalent. The same argument shows that all components of Y_{lin} of dimension d are G-equivalent to each other and that X_{lin}^{max} is G-equivalent to Y_{lin}^{max}.

Corollary 2.8. Let X, Y, Z ⊆ G be definable sets of dimension d. Assume that for every g in a large set X_{0} ⊆ X, we have dim((g + Y) ∩ Z) = d. Then X_{lin} is large in X.

Proof. Without loss of generality, X, Y, Z, X_{0} are ∅-definable, and hence every generic element g ∈ X belongs to X_{0}. For every such g ∈ X_{0} there exists c ∈ (g + Y) ∩ Z which is generic in Z over g. It follows that h = c − g is in Y and, as before, the dimension of any two of {g, h, c} is 2d. We can now find open neighborhoods V of h and W of c, defined over independent parameters, such that g + (V ∩ Y) ⊆ (W ∩ Z). Because of the genericity of g there is a neighborhood U of g such that (U ∩ X) + (V ∩ Y) ⊆ (W ∩ Z). In particular, dim((U ∩ X) + (V ∩ Y)) = d. We can therefore apply Lemma 2.7 to U ∩ X and V ∩ Y and conclude that X is locally G-linear at g.

Note that we cannot conclude, under the assumption of the last Lemma, that all components of X_{lin}^{max} are G-equivalent.

Definition 2.9. If U is an open symmetric (U = −U) neighborhood of 0 and Y ⊆ G a definable set, we say that h_{1}, h_{2} ∈ Y are U-connected in Y if h_{2} ∈ h_{1} + U and there exists a definable path in Y connecting h_{1} and h_{2}, which is contained in h_{1} + U.

Lemma 2.10. Let X be G-linear and Y a G-subset of X. Take g ∈ X, and assume that U is an open symmetric neighbourhood of 0 such that (g+U)∩X is relatively closed in g + U. Then, for any h_{1}, h_{2} ∈ Y that are U-connected in Y, we have g + (h_{1} − h_{2}) ∈ X

Proof. Because h_{1}, h_{2} are U-connected in Y, there exists a definable path γ : [a, b] → Y such that γ(a) = h_{1}, γ(b) = h_{2} and for every t ∈ [a, b], γ(t)−h_{1} ∈ U. Consider the set T of all t ∈ [a, b] such that g+(h_{1}−γ(t′)) ∈ X, for all t′ ≤ t.

We claim that T is both open and closed (with respect to the order topology in M) in [a, b]. Indeed, because (g + U) ∩ X is relatively closed in g + U, the set T is closed in [a, b].

To see that it is open, assume that t_{0} ∈ T. In particular, k = g + (h_{1} − γ(t_{0})) is in X. For t′ ∈ [a, b] close to t_{0} we have (γ(t_{0}) − γ(t′)) ∈ X − k (because ν_{γ(t_{0})}(Y)−γ(t_{0}) ⊆ ν_{k}(X)−k = ν(X) is a subgroup of G), therefore t′ ∈ T. It follows that T is open and closed in [a, b], hence T = [a, b] and therefore g + (h_{1} − h_{2}) ∈ X.

The following technical lemma ensures that we can extend every Glinear set and every G-subset of a G-linear set beyond its frontier (where the frontier of X is Cl(X) \ X).

Lemma 2.11. Assume that X is G-linear and Y is G-subset of X. Then Cl(Y) is also a G-subset of X.

Proof. Fix h in Cl(Y) \ Y. We first prove that ν_{h}(Y) − h ⊆ ν(X). Namely we show: For all h′ ∈ Y sufficiently close to h and every g ∈ X we have g + h′ − h ∈ X.

If the above fails then we have a curve γ : [a, b) → Y, with lim_{t→b} γ(t) = h such that for all t ∈ [a, b), g + γ(t) − h ∉ X. Fix U an open symmetric neighborhood of 0 such that X ∩ (g + U) is relatively closed in g + U. By choosing γ(a) sufficiently close to h, we may assume that for every t ∈ [a, b], we have γ(t) ∈ γ(a) + U (so in particular, h ∈ γ(a) + U). It follows that for every t ∈ (a, b), we have γ(a) and γ(t) are U-connected (as witnessed by γ) and therefore, by Lemma 2.10, g + γ(a) − γ(t) ∈ (g + U) ∩ X. Because X ∩ (g + U) is closed in g + U, we may take t to be b and conclude that g + γ(a) − γ(b) = g + γ(a) − h ∈ X, contradicting our assumption.

We therefore showed that for all h ∈ Cl(Y)\Y, ν_{h}(Y)−h ⊆ ν(X). By our assumption on Y this is true for every h ∈ Cl(Y). Because X is locally closed we may take the closure on the left and conclude that ν_{h}(Cl(Y))−h ⊆ ν(X) for every h ∈ Cl(Y), hence Cl(Y) is a G-subset of X.

Lemma 2.12. Let X ⊆ G be a definable set. Assume that

(i) X_{lin} is large in X (i.e. dim(X \ X_{lin}) < dim(X)) and (ii) for every h ∈ X and g ∈ X_{lin}, X − g ⊆_{0} X − h.

Then X is G-linear.

Proof. Assume that dim(X) = d. First notice that by (ii), all the components of X_{lin} are G-equivalent to each other and therefore X_{lin} is itself a G-linear set of dimension d.

Because of (ii) the local dimension of X at every point is d and therefore, by (i), X_{lin} is dense in X. By Lemma 2.11, X is a G-subset of X_{lin}.

Given any h ∈ X, we have

ν_{h}(X) − h ⊆ ν(X_{lin}) ⊆ ν_{h}(X) − h.

(the left inclusion follows from the fact that X is a G-subset of X_{lin} while the right one is just assumption (ii)).

It follows that X is locally G-linear at h hence X is G-linear.

Before the next lemma we make a small observation.

Remark 2.13. Let Z, W ⊆ G be definable sets and let f : Z → W be a definable continuous map (all are 0-definable). Then for every w ∈ f(Z) that is locally generic in W and z ∈ f^{−1}(w), we have ν_{w}(W) = f(ν_{z}(Z)). (By “locally generic” we mean that for some open V ∋ w, we have dim(w/∅) = dim(V ∩ W)).

Proof. Fix V as above and let U ⊆ G be a definable open neighborhood of z such that w is still generic in V ∩ W over the parameters (say A) defining U. Then dim(w/A) = dim(w) = dim f(U ∩ Z), w ∈ f(U ∩ Z) and therefore f(U ∩ Z) and V ∩ W have the same germ at w. In particular, ν_{w}(W) = ν_{w}(V ∩ W) ⊆ f(U ∩ Z). Because this is true for every U neighborhood of z, it follows that ν_{w}(W) ⊆ f(ν_{z}(Z)). The converse follows by continuity.

Lemma 2.14. Let X and Y be two G-linear sets. Then:

(i) X + Y is G-linear and we have

ν(X + Y) = ν(X) + ν(Y).

In particular, if X and Y are G-equivalent G-linear sets then X + Y is G-equivalent to them as well. (ii) If X and Y are G-equivalent then X ∪ Y is a G-linear set and Gequivalent to X and Y.

Proof. (i) First, notice that, by continuity, for any x ∈ X, y ∈ Y, and z = x + y we have ν_{x}(X) + ν_{y}(Y) ⊆ ν_{z}(X + Y). It follows that for all z ∈ X + Y, we have ν(X) + ν(Y) ⊆ ν_{z}(X + Y) − z.

By Remark 2.13, if z = x + y is a generic element of X + Y then ν_{z}(X + Y) − z = ν_{x}(X) + ν_{y}(Y) − z = ν(X) + ν(Y). Therefore, X + Y is G-linear at every generic z ∈ X + Y and we have ν_{z}(X + Y) − z = ν(X) + ν(Y).

It follows that (X + Y)_{lin} is large in X + Y and the germs of X + Y at all points in (X + Y)_{lin} are G-equivalent. Taken together with the above, we see that X + Y satisfies the assumptions of Lemma 2.12(1), hence it is G-linear and we have ν(X + Y) = ν(X) + ν(Y).

If X and Y are G-equivalent then ν(X) = ν(Y), hence ν(X) + ν(Y) = ν(X), so X + Y is G-equivalent to both X and Y.

(ii) It is easy to see that for all z ∈ X ∪ Y, if x ∉ F r(X) ∪ F r(Y), then we have (X ∪ Y) − z =_{0} X − z (if z ∈ X) or (X ∪ Y) − z =_{0} Y − z (if z ∈ Y). Because X and Y are G-linear and G-equivalent, it follows that (X ∪Y)_{lin} is large in X ∪ Y. Also, for every x ∈ X we clearly have X − x ⊆_{0} (X ∪ Y) − x, and similarly for y ∈ Y. We therefore can apply Lemma 2.12(1) again and conclude that X ∪ Y is G-linear, and G-equivalent to X and Y.

Lemma 2.15. Let X be a G-linear set and Y_{1}, Y_{2} two G-subsets of X. Then (i) Y_{1} + Y_{2} is a G-subset of X. (ii) Y_{1} ∪ Y_{2} is a G-subset of X.

Proof. (i) Because Y_{1} and Y_{2} are G-subsets of X, for all y_{1} ∈ Y_{1}, y_{2} ∈ Y_{2}, we have

(ν_{y_{1}}(Y_{1})+ν_{y_{2}}(Y_{2}))−(y_{1}+y_{2}) = (ν_{y_{1}}(Y_{1})−y_{1})+(ν_{y_{2}}(Y_{2})−y_{2}) ⊆ ν(X)+ν(X) = ν(X).

Again as before, if z = y_{1} +y_{2} is a locally generic element of Y_{1} +Y_{2} then

ν_{z}(Y_{1} + Y_{2}) − z = ν_{y_{1}}(Y_{1}) + ν_{y_{2}}(Y_{2}) − z ⊆ ν(X)

and hence Y_{1} + Y_{2} is G-subset of X at z. We thus have (Y_{1} + Y_{2})_{slin} dense in Y_{1} + Y_{2}. By Lemma 2.11, Y_{1} + Y_{2} is a G-subset of X.

(ii) Here we only need to note that for z ∈ Y_{1} ∩ Y_{2}, it is not true in general that the germs of Y_{1} and Y_{2} at z coincide. However, it is still true that (Y_{1} ∪ Y_{2}) − z ⊆_{0} X − g, so we can proceed as before.

We recall the following definition:

Definition 2.16. Given a definable group G in a sufficiently saturated M, a subgroup H of G is called locally-definable, if it can be written as the directed union of definable sets H = ⋃{X_{i} : i ∈ I}, where |I| < κ.

We say that H is definably connected if the X_{i}’s can all be chosen to be definably connected.

Such groups were sometimes called ⋁-definable groups (see [6]) or Inddefinable groups (see [4]). The dimension of a locally-definable group is taken to be max{dim X_{i} : i ∈ I}. Notice that if H is definably connected then it is actually definably path connected in the sense that any two points can be connected by a definable path in H. The following claim is easy to verify:

Claim 2.17. If H = ⋃{X_{i} : i ∈ I} is a locally-definable group and g ∈ H then there exists i ∈ I such that g ∈ X_{i} and

ν_{g}(H) = g + ν_{0}(H) = ν_{g}(X_{i}).

Lemma 2.18. Every locally-definable subgroup H of G can be written as the directed union of G-linear sets.

Proof. Without loss of generality all X_{i}’s in the definition of H have maximal dimension d. By Claim 2.17, for every g ∈ H there exists i ∈ I such that ν_{g}(H) = ν_{g}(X_{i}). For such an i we have: g ∈ (X_{i})_{lin}, the local dimension of X_{i} at g equals d (hence, g ∈ (X_{i})_{lin}^{max}) and all (X_{i})_{lin}^{max} are G-linear and G-equivalent. If we let X′_{i} = (X_{i})_{lin}^{max} then we have H = ⋃_{i∈I} X′_{i}.

To see that {X′_{i} : i ∈ I} is a directed system of sets: Given i, j ∈ I, we take k ∈ I such that X_{i} ∪ X_{j} ⊆ X_{k} and claim that X′_{i} ∪ X′_{j} ⊆ X′_{k}. Indeed, if g ∈ X′_{i} then ν_{g}(X′_{i}) = ν_{g}(H) and hence ν_{g}(X_{i}) = ν_{g}(X_{k}). In particular, g ∈ (X_{k})_{lin}^{max} = X′_{k}.

Lemma 2.19. We assume M is an ω-saturated structure. Let X be a definable G-linear set, 0 ∈ X. Then the following hold.

(1) The group ⟨X⟩ generated by X is locally-definable and its germ at 0 equals to ν(X). In particular, dim(⟨X⟩) = dim X.

(2) If Y a G-subset of X containing 0, then the group generated by Y is a locally-definable of dimension ≤ dim X, whose germ at 0 is contained in ν(X).

Proof. (1) The group ⟨X⟩ generated by X is a countable increasing union of the sets X_{0} = X, X_{1} = X − X, X_{2} = (X − X) + (X − X), …, so locallydefinable. By the Lemma 2.14, each X_{n} is G-linear and G-equivalent to X (so in particular, has the same dimension as X). Given g ∈ ⟨X⟩, there exists n ≥ 0 such that g ∈ X_{n}. Because of the G-linearity, for every k ≥ n, we have X_{n} =_{g} X_{k}. Because of saturation, there exists a neighborhood U of g such that U ∩ ⟨X⟩ = U ∩ X_{n}. It follows that the germ of ⟨X⟩ at this point equals to that of X_{n} and in particular, ⟨X⟩_{0} = ν(X).

(2) The group ⟨Y ⟩ generated by Y is a countable increasing union of the sets Y_{0} = Y, Y_{1} = Y − Y, Y_{2} = (Y − Y) + (Y − Y), . . .. Because −Y is also a G-subset of X, we can apply Lemma 2.15 and conclude that each Y_{n} is a G-subset of X whose germ at 0 is contained in ν(X). It follows that the dimension of ⟨Y ⟩ is at most dim X and that the germ of ⟨Y ⟩ at 0 is contained in that of ⟨X⟩.

We end this section with a small observation on locally-definable subgroups:

Lemma 2.20. Let H_{1}, H_{2} be locally-definable subgroups of G. Then H_{1}+H_{2} is a locally-definable group whose germ at 0 equals to the sum of the germs of H_{1} and H_{2} at 0.

Proof. Let

H_{1} = ⋃_{i∈I} X_{i} ; H_{2} = ⋃_{j∈J} Y_{j}.

As was pointed out above, we may assume that the X_{i}’s Y_{j}’s are all Glinear. By Lemma 2.14(1), the sets X_{i} +Y_{j} are all G-linear and ν(X_{i} +Y_{j}) = ν(X_{i}) + ν(Y_{j}). By Lemma 2.19(1),

ν_{0}(H_{1} + H_{2}) = ν(X_{i} + Y_{j}) = ν(X_{i}) + ν(Y_{j}) = ν_{0}(H_{1}) + ν_{0}(H_{2}).

3 Definable sets containing abstract subgroups

In this section M will denote an ω-saturated o-minimal expansion of an ordered group.

Theorem 3.1. Let G be a definable abelian group, Γ ⊆ G an abstract subgroup (i.e., Γ is not necessarily definable). Let X ⊆ G be a definable set containing Γ of minimal dimension d. Then there exist a definable set X′ of dimension d, a definably connected locally-definable subgroup H of dimension d, and g_{1}, …, g_{k} ∈ Γ such that Γ ⊆ X′ ⊆ ⋃_{i=1}^{k} H + g_{i}.

Proof. Assume the nontrivial case d > 0. For every g ∈ Γ, the set X∩(g+X) contains Γ and, hence by minimality, has dimension d. It follows that the set Y = {g ∈ X : dim(X ∩ X + g) = d} contains Γ and therefore, again by minimality, has dimension d.

We now apply Lemma 2.8 to Y, Y, X and X (for X_{0}, X, Y and Z, respectively), and conclude that Y_{lin} is large in Y.

Since Y contains Γ we may replace X with Y and assume from now on that X_{lin} is large in X (however, it need not be the case that X or even X_{lin} is G-linear). Moreover, we pick X ⊇ Γ such that X_{lin} has the minimal number of definably connected components of dimension d. Note that each component has infinitely many elements of Γ (otherwise, we can replace it by finitely many points). Claim 1 All the components of X_{lin} of dimension d are G-equivalent to each other. Proof Indeed, if X_{1}, X_{2} are two such components then for g ∈ X_{1} ∩ Γ, the set {h ∈ X_{2} : g + h ∈ X} = X_{2} ∩ X − g contains X_{2} ∩ Γ and therefore has dimension d (otherwise, we could replace X_{2} by a definable set of smaller dimension). For the same reason, the set of all g ∈ X_{1} such that dim(X_{2} ∩ (X − g)) = d has dimension d. Just like in the proof of Corollary 2.8, we can apply Lemma 2.7 locally to X_{1} and X_{2}, and conclude that for some open U and V we have U ∩ X_{1} is G-equivalent to V ∩ X_{2}. Because X_{1} and X_{2} are G-linear it follows that they are G-equivalent to each other. End of Claim 1.

We therefore showed that X_{lin}^{max} is G-linear. It is left to handle X^{∗} = X \ X_{lin}^{max}. Fix one of the components X_{0} of X_{lin}^{max}.

As before, for every g ∈ Γ ∩ X^{∗}, the set {h ∈ X_{0} : g + h ∈ X} contains X_{0} ∩ Γ and hence has dimension d. Therefore, after possibly replacing X^{∗} by a smaller set we may assume that for all g ∈ X^{∗} the set (g + X_{0}) ∩ X has dimension d. Claim 2 X is a G-subset of X_{lin}^{max}. Proof By abuse of notation we let ν(X) be the infinitesimal subgroup associated to the G-linear set X_{lin}^{max}. By Lemma 2.11, it is enough to see that a dense subset of X is G-linear in X_{lin}^{max}. Namely, we will show that for every locally generic g ∈ X, we have ν_{g}(X) − g ⊆ ν(X). It is clearly sufficient to consider g ∈ X^{∗}.

We fix an open set U ⊆ G and g ∈ X^{∗} which is generic in U ∩ X. By our assumption on X^{∗}, there exists h generic in X_{0} over g such that g + h (generic) in X. As in the proof of Lemma 2.7, it follows that ν_{g}(X) + h ⊆ ν_{g+h}(X), hence

ν_{g}(X) − g ⊆ ν_{g+h}(X) − (g + h) = ν(X).

(the right-most equality follows from the fact that g + h is generic in X and hence belongs to X_{lin}^{max}). End of Claim 2.

Let X_{lin}^{max} = X_{1} ∪ ⋯ ∪ X_{r} be the union of those components of X_{lin} of dimension d. By Lemma 2.4 each X_{i} is G-linear. For each X_{i}, pick g_{i} ∈ Γ ∩ X_{i}, and consider the set X′_{i} = X_{i} − g_{i}. By Claim 1, the union X′ = ⋃_{i=1}^{r} X′_{i} is G-linear (and ν(X′) = ν(X_{i}) for every i). It also contains 0 and it is definably connected. Let H′ be the subgroup of G generated by X′. By Lemma 2.19(1), dim H′ = dim X′ = d. We thus have

X_{lin}^{max} ⊆ ⋃_{i=1}^{r} H′ + g_{i}.

Let X^{1}, …, X^{t} be the definably connected components of X^{∗}. For every such X^{j}, we may assume that X^{j} ∩ Γ ≠ ∅ (for otherwise we may omit this component), take g^{j} ∈ Γ ∩ X^{j}, and let X′′ = ⋃_{i=1}^{t} X^{j} − g^{j}. By Claim 2, each X^{j} (and therefore also X^{j} − g^{j}) is a G-subset of X_{lin}^{d}. By Lemma 2.15 (2), the set X′′ is also a G-subset of X_{lin}^{max}. It is also definably connected and contains 0.

Let H′′ be the subgroup of G generated by X′′. By Lemma 2.19(2), the germ of H′′ at 0 is contained in that of X′ and therefore of H′. We now let H = H′ + H′′. By Lemma 2.20, we have ν_{0}(H) = ν_{0}(H′ + H′′) = ν_{0}(H′), so in particular the dimension of H is d.

Putting the above facts together we obtain:

Γ ⊆ X ⊆ ⋃_{i=1}^{k} H + g_{i},

for some g_{1}, …, g_{k} ∈ Γ.

With that we end the proof of the theorem.

Corollary 3.2. Let G be a definably connected abelian group. Let Γ be a divisible subgroup of the subgroup of torsion points T or(G). Let X ⊂ G be a definable set containing Γ. Then, there is a definably connected locallydefinable H subgroup of G with dim H ≤ dim X such that Γ ⊂ H.

Proof. By Theorem 3.1, taking X of minimal dimension, there is a definably connected locally-definable group H subgroup of G, with dim H = dim X, and g_{1}, …, g_{k} ∈ Γ such that Γ ⊆ ⋃_{i=1}^{k} H + g_{i}. Then Γ ⊂ H. Indeed, let g ∈ Γ and let m = lcm(o(g_{1}), …, o(g_{k})), since Γ is divisible there is h ∈ Γ such that g = mh. For such h there is an i such that h = h′ + g_{i}, for some h′ ∈ H. Then g = mh = mh′ ∈ H.

4 The main result: commutative case

In this section M will be an ω-saturated o-minimal expansion of a real closed field and G will denote a definably compact definably connected abelian group of dimension n. Such G is divisible and hence the subgroup T or(G) and any p-Sylow of G are also divisible (the p-Sylow of G is G_{p} = ⋃_{n>0} G[p^{n}], where G[m] = {g ∈ G: mg = 0_{G}}).

By Theorem 1.1 in [3] we have G[m] ≅ (ℤ/mℤ)^{n} for any m > 0, and π_{1}(G) ≅ ℤ^{n}, where π_{1}(G) is the o-minimal fundamental group of G.

Let p be a prime number and let x_{1}, …, x_{n} ∈ G[p] we say that x_{1}, …, x_{n} are n independent p-torsion points if they are 𝔽_{p}-independent under the isomorphism G[p] ≅ 𝔽_{p}^{n}.

Theorem 4.1. Let X be a definable subset of G. If X contains a p-Sylow of G, then dim X = n.

Lemma 4.2. Let p be a prime number and let x_{1}, …, x_{n} ∈ G be n independent p-torsion points. For each i = 1, …, n let τ_{i} be a path in G from 0_{G} to x_{i}, and let p τ_{i} denote the loop at 0_{G}, t ↦ p τ_{i}(t). Then, [p τ_{1}], …, [p τ_{n}] ∈ π_{1}(G) are ℤ-linearly independent.

Proof. Let φ: π_{1}(G) → G[p] the homomorphism [γ] ↦ γ̃(1), where γ̃ is the unique path in G starting at 0_{G} and such that pγ̃ = γ (see the proofs of propositions 2.10 and 2.11 in [3]). Suppose [pτ_{1}], …, [pτ_{n}] are ℤ-linearly dependent and let m_{1}[pτ_{1}] + ⋯ + m_{n}[pτ_{n}] = [k_{0_{G}}] with (m_{1}, …, m_{n}) = 1, where k_{0_{G}} is the constant loop at 0_{G}. Applying φ to this equality we get m_{1}x_{1} + ⋯ + m_{n}x_{n} = 0_{G} and hence p|(m_{1}, …, m_{n}), a contradiction.

Lemma 4.3. Let [γ_{1}], …, [γ_{n}] ∈ π_{1}(G) be ℤ-linearly independent. Then,

{∑_{i=1}^{n} γ_{i}(t_{i}): t_{i} ∈ [0, 1), 1 ≤ i ≤ n} = G.

Proof. Consider the definably compact definably connected n-dimensional abelian group 𝕋 = [0, 1)^{n} and the definable map

\begin{aligned}f: & 𝕋 & ⟶ & G \\ & (t_{1}, …, t_{n}) & ↦ & ∑_{i=1}^{n} γ_{i}(t_{i}). \end{aligned}

It suffices to prove that f is onto. Since f is continuous with respect the manifold topology of 𝕋, we can see f as a definable continuous map between definable manifolds. Claim It suffices to prove that f induces an isomorphism f_{∗}: H_{1}(𝕋; ℚ) → H_{1}(G; ℚ). Proof The map f has a degree for any orientations of 𝕋 and G, since both 𝕋 and G have dimension n (see section 4 in [3]). To prove that f is onto it suffices to prove that one (equivalently each one) of these degrees is not zero. Let ζ_{G} and ζ𝕋 be the fundamental classes of some given orientations of G and 𝕋, i.e., ζ_{G} (respectively ζ𝕋) is a generator of H_{n}(G) ≅ ℤ (resp. of H_{n}(𝕋) ≅ ℤ). The degree of f is defined by f_{∗}(ζ𝕋) = deg(f)ζ_{G}. If ω_{G} and ω𝕋 are the corresponding cohomology classes by duality we have f^{∗}(ω_{G}) = deg(f)ω𝕋. Hence to prove that deg(f) ≠ 0 suffices to prove that f induces an isomorphism of the ℚ-vector spaces f^{∗}: H^{n}(G; ℚ) → H^{n}(𝕋; ℚ). Now f^{∗}: H^{∗}(G; ℚ) → H^{∗}(𝕋; ℚ) is also ℚ-algebra homomorphism and by Theorem 1.1 in [3], both ℚ-algebras are generated by elements of degree one. So it suffices to prove that f induces an isomorphism f^{∗}: H^{1}(G; ℚ) → H^{1}(𝕋; ℚ). Applying duality again we have the required result. End of Claim.

Now, let δ_{i}: [0, 1] → 𝕋: t ↦ δ_{i}(t) = (0, …, 1^{i}, …, 0). The map f induces a map f_{∗}: π_{1}(𝕋) → π_{1}(G): [δ_{i}] ↦ f_{∗}([δ_{i}]) = [γ_{i}], for each i = 1, …, n, which is one to one and has finite cokernel (= π_{1}(G)/Im(f_{∗})) because we have n ℤ-linear independent [γ_{i}]’s. Identifying the π_{1}’s with the H_{1}’s via the Hurewicz isomorphism, we have the following exact sequence

0 → H_{1}(𝕋) →^{f_{∗}} H_{1}(G) → H_{1}(G)/Im(f_{∗}) → 0.

tensoring with ℚ we obtain that f_{∗}: H_{1}(𝕋; ℚ) → H_{1}(G; ℚ) is an isomorphism.

Corollary 4.4. Let H be a definably connected locally-definable subgroup of G. Suppose H contains G[p], for some prime p. Then H = G.

Proof. Let x_{1}, …, x_{n} be n independent p-torsion points and for each i = 1, …, n let τ_{i} be a path in H from 0_{G} to x_{i}. Let γ_{i} denote the loop at 0_{G} defined by t ↦ pτ_{i}(t), since H is a group γ_{i}([0, 1]) ⊂ H. By Lemma 4.2 [γ_{1}], …, [γ_{n}] ∈ π_{1}(G) are ℤ-linearly independent.

By Lemma 4.3, S = {∑_{i=1}^{n} γ_{i}(t_{i}): t_{i} ∈ [0, 1), 1 ≤ i ≤ n} = G. Again since H is a group S ⊂ H.

Proof of Theorem 4.1. Let X ⊆ G be a definable set containing a p-Sylow G_{p} of G. Since G_{p} is divisible, by Corollary 3.2, there exists a definably connected locally-definable group H of dimension ≤ dim X containing G_{p}. Since 𝔾_{p} contains G[p], we can apply Corollary 4.4 to get H = G and hence dim X = dim G

Corollary 4.5. Let X be a definable subset of G. If X is large in G then for any prime p and any l there is an m > l and a g ∈ X of order p^{m}.

Proof. Note that this corollary is equivalent to Theorem 4.1. Indeed, for the nontrivial case suppose there is a prime p and an l such that X ∩G_{p} ⊂ G[p^{l}]. So the set X′ = X \ G[p^{l}] is still large in G and X′ ∩ G_{p} = ∅. But then G \ X′ contains G_{p} and dim(G \ X′) < dim G, a contradiction with the theorem.

5 The main result: general case

We work in an ω-saturated structure M which is an o-minimal expansion of a real closed field M. We will use multiplicative notation for groups, and denote by p_{k} the map x ↦ x^{k}. A definably connected group which is either abelian or definably compact is divisible (see [1] or [2]), hence the map p_{k} is onto. Recall that a generic subset Y of a definable abelian group G is a definable subset of G such that finitely many translates of Y cover G.

Lemma 5.1. Let T be a definable abelian group. Suppose that for every Y ⊆ T generic, Y ∩ T or(T) ≠ ∅. Then, for each Y ⊆ T generic there is a k ∈ ℕ such that p_{k}(Y) = T.

Proof. First note that for each Y ⊆ T generic, there are g_{1}, …, g_{l} ∈ T or(T) such that T = ⋃_{i=1}^{n} g_{i}Y. Indeed, if not, by compactness, there is h ∉ T or(T)Y, i.e., hY^{−1} ∩ T or(T) = ∅, but hY^{−1} is also generic, a contradiction. Now taking k = lcm(o(g_{1}), …, o(g_{l})) we have p_{k}(Y) = p_{k}(T) = T as required.

We recall that a maximal definably connected abelian subgroup T of a definably connected definably compact group G is called a maximal torus of G and that G = ⋃_{g∈G} T^{g} (see Theorem 6.12 in [1] or [2]).

Lemma 5.2. Let G be a definably compact definably connected group. Let T be a maximal torus of G and X ⊆ G is definable. Assume the following:

(i) For every g ∈ G and for every Y ⊆ T^{g} generic, Y ∩ T or(T^{g}) ≠ ∅, and

(ii) For every g ∈ G, X ∩ T^{g} is generic in T^{g}.

Then there is a k ∈ ℕ such that p_{k}(X) = G.

Proof. Given g ∈ G, by (ii), we can apply Lemma 5.1, for Y = X ∩ T^{g}, to get a k(g) ∈ ℕ such that the map y ↦ y^{k(g)} sends X ∩T^{g} onto T^{g} (note that by divisibility of T^{g} any multiple of k(g) has the same property). Then, by compactness, there is k ∈ ℕ such that for every g ∈ G, the map y ↦ y^{k} sends X ∩T^{g} onto T^{g}. To finish the proof suffices to make use of the equality G = ⋃_{g∈G} T^{g}.

For the next lemma we will use the notions of compact domination and very good reduction introduced in [4]. Recall that a definably simple group has very good reduction and that a definably compact group which has very good reduction has also compact domination (see Theorem 10.7 in [4]).

Lemma 5.3. Let G be a definably compact definably connected group. Let X ⊆ G be a definable set such that T or(G) ⊆ X. If G has very good reduction then there is k ∈ ℕ such that p_{k}(X) = G.

Moreover, if N is a finite (central) normal subgroup of G and G/N has very good reduction then there is k ∈ ℕ such that p_{k}(X) = G.

Proof. The result can probably be read off standard Lie theory. However, since we could not find a reference we give a complete proof.

Without lost of generality we may assume that G = G(M) is defined over the reals. By our assumptions on G, G(ℝ) is a compact connected Lie group. Let T_{0} be a standard maximal torus of G(ℝ), hence T_{0} is also a Lie group and hence definable over the reals. Let T = T_{0}(M). So T is still a maximal torus of G and it has very good reduction and hence it is compactly dominated. The same happens for every conjugate T^{g} of T, for g ∈ G. Since all T^{g}’s are also abelian, we can apply Proposition 10.6 in [4], to get that for every g ∈ G, every generic subset of T^{g} has a torsion point, so condition (i) of Lemma 5.2 is satisfied.

Because T or(G) ⊆ X, the set T^{g} \ X has no torsion and hence, by (i), it is non-generic. It follows that T^{g} ∩ X is generic and therefore (ii) holds. Now we can apply Lemma 5.2 to get the required result.

For the moreover clause, let π : G → G/N be the projection map. Because N is finite, all torsion elements of G/N are in π(X), hence by what we have just proved, there is k such that p_{k}(π(X)) = G/N. If we take k′ = k · |N| then p_{k′}(X) = G (because N is central and G divisible).

Notice, in the setting of the above lemma, that if p_{k}(X) = G then in particular, dim X = dim G (since the image of X under a definable map cannot increase).

We can now prove the main result:

Theorem 5.4. Let G be a definably compact group and assume that X ⊆ G is a definable set containing all torsion points of G. Then dim X = dim G.

Proof. By 4.1 in [5] and 5.4 in [8], after modding out G_{1} = G/Z(G) by its finite center, the group we obtain is a direct product of simple groups. In particular, it has very good reduction (see the proof of 5.1 in [7]) and hence the “moreover” clause of Lemma 5.3 holds for G/Z(G). We therefore have:

(i) For every definable X_{1} ⊆ Z(G), if T or(Z(G)) ⊆ X_{1} then dim X_{1} = dim Z(G) (by Theorem 4.1 this is true for subsets of Z(G)^{0}, but then it clearly follows for subsets of Z(G) as well).

(ii) For every definable X_{1} ⊆ G, if T or(G/Z(G)) ⊆ X_{1}/Z(G) then dim(X_{1}/Z(G)) = dim(G/Z(G)). (by Lemma 5.3 and our previous observation).

We now proceed as follows, with X ⊆ G a definable set containing T or(G). Given g ∈ T or(G) and h ∈ T or(Z(G)), we have gh ∈ T or(G) ⊆ X. Hence, for every such g, we have

X_{g} = {h ∈ Z(G) : gh ∈ X} ⊇ T or(Z(G)),

so, by (i), dim X_{g} = dim Z(G). Since gX_{g} = (gZ(G)) ∩ X, we have, for every g ∈ T or(G),

dim gZ(G) ∩ X = dim Z(G).

Let

X_{1} = {g ∈ G : dim(gZ(G) ∩ X) = dim Z(G)}.

It follows from the above that T or(G) ⊆ X_{1}, and therefore T or(G/Z(G)) ⊆ X_{1}/Z(G). (Indeed, if hZ(G) ∈ T or(G/Z(G)) then for some n ∈ ℕ, h^{n} ∈ Z(G), and therefore for some k ≥ n, h^{k} ∈ Z(G)^{0}. Because Z(G)^{0} is divisible there exists h_{1} ∈ Z(G), with h_{1}^{k} = h^{k}. Because h_{1} is central, (hh_{1}^{−1})^{k} = e, hence hh_{1}^{−1} ∈ X_{1} and therefore hZ(G) ∈ X_{1}/Z(G)). By (ii), dim(X_{1}/Z(G)) = dim(G/Z(G)).

To finish the proof, consider the set Y = (X_{1}Z(G)) ∩ X. The cosets of Z(G) partition Y into equivalence classes, each of dimension dim Z(G) (by definition of X_{1}). Since every Z(G)-coset of an element in X_{1} intersects X nontrivially, we have

dim(Y/Z(G)) = dim(X_{1}/Z(G)) = dim(G/Z(G)).

Summarizing, we have

dim Y ≥ dim Z(G) + dim(G/Z(G)) = dim G,

and hence (since Y ⊆ X) dim X = dim G.

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