虬江文库021 | Anand Pillay，姚宁远：p-进数域上的可定义群

Abstract：It is known Hrushovski and Pillay (Israel J Math 85:203–262, 1994) that a group G definable in the field Q_p of p-adic numbers is definably locally isomorphic to the group H(Q_p) of p-adic points of a (connected) algebraic group H over Q_p. We observe here that if H is commutative then G is commutative-by-finite. This shows in particular that any one-dimensional group G definable in Q_p is commutative-by-finite. This result extends to groups definable in p-adically closed fields. We prove our results in the more general context of geometric structures.

姚宁远：复旦大学哲学学院副教授，中山大学工学博士，研究方向为模型论NIP(非独立性质)理论，O-minimal理论，p-adic群，可定义顺从群。

本文为姚宁远与Anand Pillay（University of Notre Dame, USA）合作撰写，发表于Archive for Mathematical Logic （2019）58，pp. 1029-1034。

We here consider questions about the p-adic field (or more generally p-adically closed fields) which are analogous to questions long studied for the real field (more generally real closed fields, o-minimal structures), namely the classification and description of definable groups. (We are not claiming that such a classification has been already achieved in the real case, just that it has been studied for a long time.) We emphasize definable rather than interpretable, which happen to coincide in the real case because of elimination of imaginaries, but not in the p-adic case. Common features are that definable groups have naturally the structure of real and p-adic Lie groups [6,7], are locally isomorphic to real and p-adic algebraic groups [2], and the definable fields are classified [7]. In both the real and p-adic contexts, definable sets have a dimension which has a topological description as well as an algebraic description (the algebro-geometric dimension of the Zariski closure). However there are big differences when it comes to questions of (definable) connectedness. In the real case, we have the descending chain condition on definable subgroups, definable groups are (definably) connected-by-finite, and if H is a definable subgroup of G of infinite index in G then dim(H) < dim(G). It is well-known that in the p-adic context these properties fail drastically. 

From the facts described above, in the real case, if G is a definable group and has a definable subset X with dim(X) = dim(G) and all elements of X commute with each other, one concludes that G is commutative-by-finite: the centre of the centralizer of X is a commutative definable subgroup of G which contains X so has the same dimension as G, hence has finite index. Although this proof does not work in the p-adic context, we will show that the result still holds. In fact, we situate the result in the more general context of groups definable in geometric structures (in the sense of Section 2 of [2]). Our proof is relatively soft.

The referee has told us that Shelah has a result which is related to ours and which should be mentioned in the paper. From what the referee said, we assume this result of Shelah to be Claim 4.3 of [8], which produces an infinite definable commutative subgroup H of a definable group G from an infinite (not necessarily definable) set X of pairwise commuting elements ofG, under the assumption that the ambient theory T is NIP. If T is, in addition, a geometric theory, and X is definable with dim(X) = k, then Shelah’s proof should give H with dim(H) ≥ k (so of maximal dimension if X has maximal dimension). 

Actually, in our p-adic application (over the standard model Q_p), owing to the fact that a neighbourhood basis of the identity of a definable group G consists of open definable subgroups, one sees that if G is definably locally isomorphic to a commutative algebraic group H, then already G has a definable commutative subgroup which is open, so of maximal dimension. So, in this p-adic situation, we already have the conclusion of Shelah’s result. Themain new contribution of this paper is producing a definable commutative subgroup of finite index. As remarked in the abstract we conclude that any one-dimensional group definable in the p-adics is commutative-by-finite. 

A commutative-by-finite group G is amenable (as a discrete group), hence definably amenable with respect to any ambient structure in which G happens to be definable. As Th(Q_p) is NIP (see Section 4.2, [1]), results from [4] which refine the “algebraic group configuration theorem” of [2] will apply to G. So even though the current paper is short we thought it makes sense as a stand-alone paper as it is self-contained, the methods are elementary, and it may be useful for future work. 

We use fairly basic model theory.We refer to the excellent survey [1] as wel as [2,5] for the model theory of the p-adic field (Q_p,+,×, 0, 1). In fact both [2,5] are also good references for the model theoretic background required for the current paper. A geometric structure (see Section 2 of [2]) is a one-sorted structure M such that in any model N of Th(M), algebraic closure satisfies exchange (so gives a so-called pregeometry on N), and there is a finite bound on the sizes of finite sets in uniformly definable families. The structure (Q_p,+,×, 0, 1) is an example of a geometric structure ([2], Proposition 2.11), as model-theoretic algebraic closure coincides with field-theoretic algebraic closure. Let us note, for the referee’s sake, that we have not so far mentioned or talked about so-called geometric fields. In fact we do not need to mention them for the main results of this paper. However they are mentioned in Question 2.6 (ii) below, and the reader can see 2.9 in [2] for the definition. 

In a geometric structure M, if a is a finite tuple from M and B a subset of M then dim(a/B) denotes the size of a maximal subtuple of a which is algebraically independent over B. If M is sufficiently saturated and X is a B-definable subset of Mⁿ (where B is finite) then dim(X) = max{dim(a/B) : a ∈ X}. It is important to know that when M is (Q_p,+,×, 0, 1), and X ⊆ Mⁿ is definable, then its dimension in sense of a geometric structure coincides with its “topological dimension”, namely the greatest k ≤ n such that the image of X under some projection from Mⁿ to M^k contains an open set. 

In one of the general results below we make an assumption on the existence of G⁰. So we explain what this means, although it is discussed in the first section of [5]. Let M be a model, A a set of parameters from M, and G a group definable over A in M. We assume that M is |A|+|L|-saturated, where L is the cardinality of the language, always assumed, notationally, to be infinite. Then by G⁰_A we mean the intersection of all A-definable subgroups of G of finite index. We say that G⁰ exists if G⁰_A does not depend on A, namely cannot get smaller by increasing A, and increasing the model M. Note that the existence of G⁰ is equivalent to the nonexistence of an infinite uniformly definable family of subgroups of G of some fixed finite index, which amounts to the DCC on intersections of uniformly definable subgroups of (a given) finite index. As mentioned in [5], the existence of G⁰ follows from the ambient theory having NIP. 

Finally, let us state clearly the local isomorphism results alluded to earlier.

Fact 1.1 [7] Let G be a group definable in the field Q_p. Then G has definably the structure ofa p-adic Lie group. Moreover ifG has dimension k as a definable group then it has dimension k as a p-adic Lie group.

Fact 1.2 [2] Suppose G is a group definable in the field Q_p. Consider G with its topology given by Fact 1.1. Then there is a connected algebraic group H over Q_p, such that the algebraic-geometric dimension of H equals the dimension of G, and there is a definable homeomorphism f between an open neigbourhood U ofthe identity in G and an open neighbourhood V of the identity of(the p-adic Lie group) H(Q_p) such that f(ab) = f(a) f(b) whenever a, b ∈ U and ab ∈ U.

Remark 1.3 (i) In Fact 1.2 we can choose f to be a definable isomorphism (and homeomorphism) between definable open subgroups of G and H(Q_p), because any (definable) p-adic Lie group has a (definable) compact open subgroup, and a compact p-adic Lie group is profinite.

(ii) Both Fact 1.1 and Fact 1.2 hold (with appropriate definitions of definable Lie group over a p-adically closed field K) for groups G definable in an elementary extension K of Q_p. See Step 1 of the proof of Theorem 2.1 in [3] in the real closed field situation which works word for word in the p-adically closed field case. On the other hand, part (i) of this remark really requires working in the standard model Q_p.

We start with an easy lemma about geometric structures. As the referee points out, this is well-known and connected with geometric structures being “real rosy”.

Lemma 2.1 Suppose M is a geometric structure, X ⊆ Mⁿ is a definable set of dimension k, and f is a definable function from X to M^m. Suppose that f⁻¹(b) has dimension k for all b ∈ Im(f) then Im(f) is finite.

Proof We may assume M to be saturated and work over the parameters over which X and f are defined. Suppose for a contradiction Im(f) to be infinite. Then we can find b ∈ Im(f) such that dim(b) ≥ 1. As dim(f⁻¹(b)) = k we can find a ∈ f⁻¹(b) such that dim(a/b) = k. Hence by subadditivity, dim(a, b)> k. As b ∈ dcl(a) it follows that dim(a)> k, contradicting that a ∈ X and dim(X) = k.

Remark 2.2 The conclusion of the lemma is weaker than stating that there is no definable equivalence relation on a definable k-dimensional set with infinitely many classes of dimension k. The latter is false in Q_p.

Proposition 2.3 Let M be a geometric structure, and let G ⊆ Mⁿ be a group definable in M with dim(G) = k. Assume that (working in a saturatedmodel) G⁰ exists. Suppose that G contains a definable subset X ofdimension k such that C_G(a) has dimension k for all a ∈ X. Then G is commutative-by-finite, namely G has a (definable) subgroup H offinite index such that H is commutative.

Proof We begin with:

Claim Let Y be an arbitrary (not necessarily definable) subset of G such that C_G(a) has dimension k for all a ∈ Y. Then C_G(Y) is a definable subgroup of G of finite index.

Proof of Claim Let a ∈ Y and let f_a be the function from G to G defined by f_a(g) = gag⁻¹a⁻¹. If g1, g2 ∈ G, then g1ag₁^-1a⁻¹ = g2ag₂⁻¹ iff g₂^-1g1ag₁^-1g2 = a, namely g1g₂^-1 ∈ C_G(a). So

(*) f_a(g1) = f_a(g2) iff g₁C_G(a) = g₂C_G(a). 

So for each h ∈ Im(f_a), f_a⁻¹(h) is a translate of C_G(a) which by hypothesis has dimension k. By Lemma 2.1, Im(f_a) ⊆ G is finite. As the Im(f_a) are uniformly definable finite subsets of G, there is finite bound on their size. By (*) Im(f_a) is in bijection with G/C_G(a), hence there is a finite bound on the index ofC_G(a)) in G (for a ∈ Y). Our assumption that G⁰ exists (as discussed in the previous section) implies that C_G(Y) =∩_a∈YC_G(a) is a finite subintersection, so is a definable subgroup G, of finite index. The proof of the claim is complete. 

We first apply the claim with Y = X, to see that C_G(X) is a definable subgroup of

G of finite index. Let H = C_G(X). Now note that the claim also applies if we take Y = H (as for every a ∈ H, C_G(a) contains X so has dimension k). So C_G(H) has finite index in G. But then H ∩C_G(H) also has finite index in G and is commutative. This concludes the proof of the proposition.

We conclude:

Theorem 2.4 (i) Let G be a group definable in Q_p. Let H be a connected algebraic group over Qp as in Fact 1.2. Suppose that H is commutative, then G is commutative-by-finite.

(ii) Let G be a group of dimension 1 definable in Q_p. Then G is commutative-by-finite.

Proof (i) Let U, V be definable open neighbourhoods of the identity of G, H(Q_p) respectively and f : U → V as given by Fact 1.2. By choosing a smaller definable open neighbourhood U₁ of the identity contained in G such that ab ∈ U for a, b ∈ U₁, we see from the assumptions that ab = ba for a, b ∈ U₁. Now the p-adic topological dimension of U₁ coincides with that of G, but both coincide with the dimension with respect to Q_p as a geometric structure. So, bearing in mind that G⁰ exists (as remarked in the introduction), we can apply Proposition 2.3 to conclude that G is commutative-by-finite.

(ii) If G has dimension 1, then by Fact 1.2 the connected algebraic group H has dimension 1 as an algebraic group, so is commutative. So part (i) implies.

Remark 2.5 By Remark 1.3 (ii), Theorem 2.4 goes through for groups definable in arbitrary p-adically closed fields (i.e. models of Th(Q_p)).

Question 2.6 (i) Do we need the assumption that G⁰ exists in Proposition 2.3? We presume yes, namely there is a counterexample without it.

(ii) Let F be a geometric field in the sense ofDefinition 2.9 of [2]. Let G be a group definable in F and let H be a connected algebraic group over F given by Proposition 3.1 of [2]. Suppose H is commutative. Can one find a definable subset X of G of dimension equal to dim(G) such that for all a ∈ X, C_G(a) has dimension equal to dim(G)?

Acknowledgements Both authors would like to thank the Institut Henri Poincaré, Paris, for its hospitality and support during the trimester on model theory in early 2018 when this work was done. The second author would like to thank the IHES, Orsay, for its hospitality during the academic year 2017–2018. Both authors would like to thank Immi Halupczok for discussions. Finally, both authors would like to thank the referee for his or her comments which led to some positive changes in the paper. The introduction has been expanded to include comparisons with earlier results, and the proof of the main result, Proposition 2.3, has been improved.

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