On non-compact $p$-adic definable groups

On non-compact p-adic definable groups Will Johnson and Ningyuan Yao arXiv:2103.12427v2 [math.LO] 18 May 2022 May 19, 2022 Abstract In [16], Peterzil and Steinhorn proved that if a group G definable in an ominimal structure is not definably compact, then G contains a definable torsionfree subgroup of dimension one. We prove here a p-adic analogue of the PeterzilSteinhorn theorem, in the special case of abelian groups. Let G be an abelian group definable in a p-adically closed field M . If G is not definably compact then there is a definable subgroup H of dimension one which is not definably compact. In a future paper we will generalize this to non-abelian G. 1 Introduction In [16], Peterzil and Steinhorn prove that if G is a definable group in an o-minimal structure M, and G is not definably compact, then G has a definable 1-dimensional subgroup H that is not definably compact. To prove this, they take a continuous unbounded definable curve I : [0, +∞) → G and take H to be the “tangent line at ∞.” This can be made precise using the language of µ-types and µ-stabilizers developed later by Peterzil and Starchenko [15]. Say that two complete types q, r ∈ SG (M) are “infinitesimally close” if there are realizations a |= q and b |= r such that ab−1 is infinitesimally close to idG (that is, ab−1 is contained in every M-definable neighborhood of idG ). This is an equivalence relation on SG (M), and equivalence classes are called “µ-types.” The “µ-stabilizer” stabµ (q) of q ∈ SG (M) is the stabilizer of the µ-type of q. With these definitions, the “tangent line of I at ∞” is simply the µ-stabilizer of the type on I at infinity, an unbounded 1-dimensional definable type. (Here, we say that a type q ∈ SG (M) is “unbounded” if no formula in q defines a definably compact subset of G.) Peterzil and Steinhorn essentially show that the µ-stabilizer of an unbounded 1-dimensional definable type is a torsion free non-compact definable subgroup of dimension 1. More generally, in [15], Peterzil and Starchenko consider a general definable type q ∈ SG (M), showing that stabµ (q) is a torsion-free definable group of a certain dimension. It is natural to ask whether analogous results hold in the theory pCF (p-adically closed fields). There are many formal similarities between pCF and o-minimal theories, especially RCF (real closed fields). In both settings, definable groups can be regarded as real or p-adic Lie groups [17, 18], and are locally isomorphic to real or p-adic algebraic 1 groups [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). On the other hand, definable connectedness behaves very differently in the two settings. In this paper, we restrict our attention to one-dimensional definable types, as in the original work of Peterzil and Steinhorn [16]. Unfortunately, we must also assume that G is “nearly abelian” for most of our theorems. Definition 1.1. Let G be a definable group in a model of pCF. G is nearly abelian if there is a definably compact definable normal subgroup K ⊆ G with G/K abelian. See Definition 2.1 for a precise definition of “definable compactness,” and Propositions 2.16 and 2.24 for some equivalent conditions. Our main results are as follows: Theorem 1.2. Let G be a definable group over a p-adically closed field M. If G is not definably compact and G is nearly abelian, then there is a 1-dimensional definable subgroup H ⊆ G that is not definably compact. We plan to generalize Theorem 1.2 to non-abelian groups in a future paper. Theorem 1.3. Suppose that G is a definable group over an ℵ1 -saturated p-adically closed field M. Then for any definable unbounded 1-dimensional type r ∈ SG (M), the µ-stabilizer stabµ (r) is a 1-dimensional type-definable subgroup of G. If G is abelian (or nearly abelian), then stabµ (r) is unbounded. Here, a set or type is “bounded” if it is contained in a definably compact set, and “unbounded” otherwise (Definition 2.9). The assumption on saturation is necessary. For example, suppose M = Qp , G is the multiplicative group, and r ∈ SG (Qp ) is one of the definable types consistent with {x | v(x) < Z}. Then stabµ (r) is the intersection of all n-th powers Pn = {x | x 6= 0 ∧ ∃(x = y n )}, which is the trivial group {1}. We can also say something when M is not saturated, but we will need a few more definitions from [15]. Fix a group G definable in a p-adically closed field M. For any partial type Σ(x) in G, and any L-formula φ(x; y), let stabφ (Σ) denote \ stab{g ∈ G(M) | Σ ⊢ φ(gx; b)}. b∈M k (This can be understood as T the stabilizer of the φ(z · x; y)-type generated by Σ(x).) It turns out that stab(Σ) = φ∈L stabφ (Σ). Now suppose that r is a type in SG (M). Let µ be the partial type of “infinitesimals,” that is, the set of LM -formulas definining neighborhoods of idG . Let µ · r be the partial type such that (µ · r)(N) = µ(N) · r(N) for sufficiently saturated N ≻ M. It turns out that \ stabµ (r) = stab(µ · r) = stabφ (µ · r). φ∈L Moreover, when r is definable, the groups stabφ (µ · r) are definable, and stab(µ · r) is type-definable. (This is the reason why stabµ (r) is type-definable in Theorem 1.3. In the o-minimal case, there is a descending chain condition on definable groups, which ensures that stabµ (r) is definable in [15].) 2 Theorem 1.4. Suppose that G is a definable group over a p-adically closed field M. Let r ∈ SG (M) be a definable unbounded 1-dimensional type. Then there is a formula φ ∈ L such that stabφ (µ · r) is a 1-dimensional definable subgroup of G. When G is abelian (or nearly abelian), stabφ (µ · r) is unbounded. Our proofs of these theorems are based on the original proofs of Peterzil and Steinhorn [16], though several important changes are necessary. First of all, the µ-stabilizer stabµ (r) is no longer definable, but merely type-definable, as mentioned above. For this reason, it is necessary to compute the stabilizers in an |M|+ -saturated elementary extension N ≻ M. A more serious problem arises when trying to generalize [16, Lemma 3.8]. This lemma, which is used to show that stabµ (p) 6= {idG }, roughly says the following: if I is a curve tending to infinity and B is an annulus around idG , then g · B ∩ I 6= ∅ for all g ∈ I. This follows by a simple connectedness argument (I is connected, so it must cut across the annulus g·B on its way from g to infinity). This argument fails critically in the totally disconnected p-adic context. In Section 4 we develop an alternative argument to replace [16, Lemma 3.8]. Unfortunately, the argument only works properly in the abelian (or near-abelian) case. 1.1 Notation and conventions We shall assume a basic knowledge of model theory, including basic notions such as definable types, saturation, heirs, and so on. Good references are [12, 21]. We refer to the excellent survey [1] as well as [13, 7] for the model theory of the p-adic field (Qp , +, ×, 0, 1). In fact, [13] and [7] are also good references for the model theoretic background required for the current paper. Let T be a theory in some language L. We write M for a monster model of T , in which every type over a small subset A ⊆ M is realized, where “small” means |A| < κ for some big enough cardinal κ. The letters M, N, M ′ and N ′ will denote small elementary submodels of M. We will use x, y, z to mean arbitrary n-tuples of variables and a, b, c ∈ M to denote n-tuples in Mn with n ∈ N. Every formula is an LM -formula. For an LM -formula φ(x), φ(M) denotes the definable subset of M |x| defined by φ, and a set X ⊆ M n is definable if there is an LM -formula φ(x) such that X = φ(M). If M ≺ N ≺ M, and X ⊆ N n is defined by a formula ψ with parameters from M, then X(M) and X(M) will denote ψ(M) and ψ(M) respectively; these are clearly definable subsets of M n and Mn respectively. Following [15, Definition 2.12], we say that a partial type Σ is A-definable or definable over A if for every formula φ(x; y), there is an LA -formula ψ(y) such that Σ(x) ⊢ φ(x; b) ⇐⇒ M |= ψ(b) for all b ∈ M. We will denote the formula ψ(y) by (dΣ x)φ(x, y), thinking of dΣ as a quantifier. The map φ(x; y) 7→ (dΣ x)φ(x, y) is called the definition schema of Σ(x). If Σ(x) is a definable partial type over M, and N ≻ M, then ΣN will denote the canonical extension of Σ by definitions, i.e., the following partial type over N: ΣN = {φ(x; a) ∈ LN | N |= (dΣ x)φ(x, a)}. 3 When p is a complete definable type over M, the canonical extension pN is the same thing as the unique heir of p over N. For a definable set D ⊆ M n , and φ(x) an LM -formula, we say that φ(x) is a Dformula if M |= φ(x) =⇒ x ∈ D(M). A partial type q(x) (over a small subset) is a D-type if q(x) ⊢ x ∈ D(M). We write SD (M) for the space of complete D-types over M. We consider Qp as a structure in the language of rings L = Lr = {+, ×, −, 0, 1}. The valuation ring Zp is definable in Qp . The valuation group (Z, +, <) and the valuation v : Qp → Z ∪ {∞} are interpretable. A p-adically closed field is a model of pCF := Th(Qp ). For any M |= pCF, R(M) will denote the valuation ring, and ΓM will denote the value group. By [11], pCF admits quantifier elimination after adjoining predicates Pn for the n-th power of the multiplicative group for all n ∈ N+ . The theory pCF also has definable Skolem functions [2]. The p-adic field Qp is a locally compact topological field, with basis given by the sets B(a, n) = {x ∈ Qp | x 6= a ∧ v(x − a) ≥ n} for a ∈ Qp and n ∈ Z. The valuation ring Zp is compact. The topology is definable (as in Section 2.1 below), so it extends to any p-adically closed field M, making M a topological field (usually not locally compact). Any definable set X ⊆ M n has a topological dimension, denoted by dim(X), which is the maximal k ≤ n such that the image of the projection π : X → M n ; (x1 , . . . , xn ) 7→ (xr1 , . . . , xrk ) has interior, for suitable 1 ≤ r1 < · · · < rk ≤ n. As model theoretic algebraic closure coincides with the field-theoretic algebraic closure, algebraic closure gives a pregeometry on M, and the algebraic dimension dimalg (X) of X can be calculated in the usual way. The topological dimension coincides with the algebraic dimension. 1.2 Outline In Section 2, we review the notion of definable compactness, and how it behaves in definable manifolds and definable groups in pCF. In Section 3 we review the theory of dp-rank, which is used in Section 4. In Section 4, we prove a technical statement about “gaps” in unbounded sets, which replaces the use of connectedness in Peterzil-Steinhorn [16, Lemma 3.8]. In Section 5, we review the theory of stabilizers and µ-stabilizers from [15]. Finally, we prove the main theorems in Section 6. 2 Definable compactness In this section, we review the notion of definable compactness for definable manifolds and definable groups in p-adically closed fields. The treatment of (p-adic) definable compactness in the literature is questionable, so we build up the theory from scratch, out of an abundance of caution. In Section 2.1 we recall an abstract definition of definable compactness, which behaves well in any definable topological space. In the next two sections, we restrict our attention to p-adic definable manifolds. In Section 2.2 we show that our definition agrees 4 with the definition in the literature in terms of curve completion. In Section 2.3 we give another characterization using specialization of definable types. Finally, in Section 2.4 we list some consequences for definable groups. 2.1 Abstract definable compactness Let M be an arbitrary structure. A definable topology on a definable set X ⊆ M n is a topology with a (uniformly) definable basis of opens. A definable topological space is a definable set with a definable topology. Recall that a topological space is compact if any filtered intersection of non-empty closed sets is non-empty. Definition 2.1. Let X be a definable topological space in a structure M. Say that X is definably compact if the following holds: for any definable family T F = {Yt : t ∈ T } of non-empty closed sets Yt ⊆ X, if F is downwards directed, then F 6= ∅. More generally, say that a definable set Y ⊆ X is definably compact if it is definably compact with respect to the induced subspace topology. Definable compactness has many of the expected properties: Fact 2.2. 1. If X is a compact definable topological space, then X is definably compact. 2. If X, Y are definably compact, then X × Y is definably compact. 3. If f : X → Y is definable and continuous, and X is definably compact, then the image f (X) ⊆ Y is definably compact. 4. If X is a Hausdorff definable topological space and Y ⊆ X is definably compact, then Y is closed. 5. If X is definably compact and Y ⊆ X is closed and definable, then Y is definably compact. 6. If X is a definable topological space and Y1 , Y2 ⊆ X are definably compact, then Y1 ∪ Y2 is definably compact. Definition 2.1 and Fact 2.2 are due independently to Fornasiero [4] and the first author [9, Section 3.1]. Remark 2.3. Suppose X is a definable topological space in a structure M, and N ≻ M. Then X(N) is naturally a definable topological space in the structure N, and X(N) is definably compact if and only if X is definably compact. In other words, definable compactness is invariant in elementary extensions. 5 2.2 Definable compactness and definable manifolds in pCF Let M be a p-adically closed field with valuation group ΓM . Each power M n is a definable topological space. We first characterize definable compactness for subsets of M n . Lemma 2.4. If X ⊆ M n is definably compact, then X is closed and bounded. Proof. For t ∈ M \ {0}, let Ot be the n-dimensional ball B(0, v(t))n . Each Ot is clopen in M n . Therefore {X \ Ot : t ∈ M \ {0}} is a downwards-directed definable family of closed subsets of X, with empty intersection. By definable compactness, there is some t such that X \ Ot = ∅, or equivalently, X ⊆ Ot . Then X is bounded. Closedness follows similarly, or by Fact 2.2(4). Lemma 2.5. If X ⊆ M n is closed and bounded, then X is definably compact. Proof. Equivalently, if {Yt } T is a downwards-directed definable family of non-empty, closed, bounded sets, then t Yt 6= ∅. This claim can be expressed as a countable conjunction of L-sentences. (We need infinitely many sentences because there is no bound on the complexity of the definable family {Yt }.) As a countable conjunction of L-sentences, the claim holds in M if and only if it holds in Qp . Therefore, we may assume that M = Qp . In this case, the set X will be compact, and hence definably compact by Fact 2.2(1). Definition 2.6. Let X be a definable topological space. A Γ-exhaustion is a definable family {Wγ | γ ∈ ΓM } such that • Each Wγ is an open, definably compact subset of X. In particular, Wγ is clopen. • If γ ≤ γ ′ , then Wγ ⊆ Wγ ′ . S • X = γ∈ΓM Wγ . Lemma 2.7. If U ⊆ M n is definable and open, then U has a Γ-exhaustion. Proof. For any x̄ = (x1 , . . . , xn ) Q ∈ M n and γ ∈ ΓM , let B(x̄, γ) denote the ball of valuative radius γ around x̄, i.e., ni=1 B(xi , γ). Let Wγ be the set of x ∈ U such that B(x, γ) ⊆ U and 0̄ ∈ B(x, −γ). We claim that the family Wγ is a Γ-exhaustion. First of all, for all x′ sufficiently close to x, we have B(x, γ) = B(x′ , γ) and B(x, −γ) = B(x′ , −γ), and so x ∈ Wγ ⇐⇒ x′ ∈ Wγ . Therefore Wγ is clopen. Additionally, x ∈ Wγ =⇒ 0̄ ∈ B(x, −γ) ⇐⇒ x ∈ B(0̄, −γ). Therefore Wγ is bounded. By Lemma 2.5, Wγ is definably compact. If γ ′ ≥ γ, then B(x, γ ′ ) ⊆ B(x, γ) and B(x, −γ ′ ) ⊇ B(x, −γ). Therefore x ∈ Wγ =⇒ x ∈ Wγ ′ , and the family {Wγ } is monotone. Lastly, if x ∈ U, then for sufficiently large γ, we have B(x, γ) ⊆ U, because U is open. Also, 0̄ ∈ B(x, S−γ) for sufficiently large γ. Thus x ∈ Wγ for all sufficiently large γ. This shows U = γ Wγ . 6 An n-dimensional definable manifold over M is a Hausdorff definable topological space X with a covering by finitely may open subsets U1 ,. . . ,Um , and a definable homeomorphism from Ui to an open set Vi ⊆ M n for each i. Proposition 2.8. Let X be a definable manifold in M. Then X has a Γ-exhaustion. n Proof. Cover X with finitely many open sets Ui homeomorphic Sto open subsets of M . For each i, let {Wi,γ }γ∈ΓM be a Γ-exhaustion of Ui . Let Vγ = i Wi,γ . Then the family {Vγ } is a Γ-exhaustion of X. Definition 2.9. Let X be a definable manifold. An arbitrary subset Y ⊆ X is bounded if Y ⊆ D for some definably compact subset D ⊆ X. Proposition 2.10(1) gives a more concrete definition of “bounded” in terms of Γexhaustions. Proposition 2.10. Let X be a definable manifold and Y ⊆ X be an arbitrary subset. 1. Let {Wγ } be a Γ-exhaustion of X. Then Y is bounded if and only if there is γ ∈ Γ such that Y ⊆ Wγ . 2. Suppose Y is definable. Then Y is definably compact if and only if Y is closed and bounded. 3. Suppose Y is definable. Then Y is bounded if and only if the closure Y is definably compact. Proof. 1. If Y ⊆ Wγ , then Y is contained in the definably compact set Wγ . Conversely, suppose Y is bounded, witnessed by a definably compact set Z ⊆ X with Y ⊆ Z. The filtered intersection \ (Z \ Wγ ) γ is empty, so there is some γ such that Wγ ⊇ Z ⊇ Y . 2. If Y is definably compact, then Y is closed (Fact 2.2(4), and Y is bounded because Y ⊆ Y . Conversely, suppose that Y is closed and bounded. Then Y is a definable closed subset of a definably compact set, so Y is definably compact by Fact 2.2(5). 3. If Y is definably compact, then Y is bounded because Y ⊆ Y . Conversely, suppose that Y is bounded. Then Y ⊆ Z for some definably compact set Z ⊆ X. The cosure Y is a definable closed subset of Z, so Y is definably compact by Fact 2.2(5). Remark 2.11. Definable compactness is a definable property: Let Xt be a definable manifold depending definably on some parameter t ∈ T . Then {t ∈ T : Xt is definably compact} is definable. This can be proved from Proposition 2.10(1,2) by compactness, using Remark 2.3 to reduce to the case where M is highly saturated. 7 Remark 2.12. When M = Qp , a definable manifold X is definably compact if and only if it is compact. One direction is Fact 2.2(1). Conversely, suppose X is definably compact. Cover X by definable open subsets U1 , . . . , Un , each homeomorphic to an open subset of M n . S As in the proof of Proposition 2.8, let {Wi,γ }γ∈Z be a Γ-exhaustion of Ui , and let Vγ = ni=1 Wi,γ , so that {Vγ }γ∈Z is a Γ-exhaustion Sn of X. By Proposition 2.10, there is some γ ∈ Z such that X = Vγ . Then X = i=1 Wi,γ , where each Wi,γ is definably compact. Lemmas 2.4 and 2.5 imply that definable compactness is equivalent to compactness for definable subsets of M n . Therefore each Wi,γ is compact. As X is covered by finitely many compact sets, X itself is compact. We now try to relate our notion of definable compactness to the more familiar notions appearing in [13]. Definition 2.13. Let X be a definable manifold. Let D be a definable subset of M \{0} with 0 ∈ D. Let f : D → X be a definable function. Then a ∈ X is a cluster point of f if (0, a) is in the closure of the graph of f . In other words, for every neighborhood U1 of 0 and every neighborhood U2 of a, there is x ∈ U1 ∩ D such that f (x) ∈ U2 . Lemma 2.14. let X be a definable manifold. Let f : R(M) \ {0} → X be a definable function. Then f is continuous at all but finitely many points of R(M). Proof. An exercise using the fact that any definable function M → M n is continuous off a finite set. Lemma 2.15. Let X be a definable manifold. Let Y be a definable subset. The following are equivalent: 1. Y is definably compact. 2. If D is a definable subset of M \ {0} with 0 ∈ D, then every definable function f : D → Y has a cluster point. 3. Any definable continuous function f : R(M) \ {0} → Y has a cluster point in Y . 4. Any definable continuous function f : B(0, γ) \ {0} → Y has a cluster point in Y . 5. Let {Zγ }γ∈ΓM be a definable T family of non-empty closed subsets of Y , such that γ ≤ γ ′ =⇒ Zγ ⊇ Zγ ′ . Then γ∈ΓM Zγ 6= ∅. Proof. (1)⇒(2): the set of cluster points is the intersection \ f (B(0, γ) ∩ D). γ∈ΓM This is non-empty by definable compactness of Y . (2)⇒(3) is trivial, and (3)⇒(4) follows by rescaling. (4)⇒(5): By definable Skolem functions, there is some definable function f : M \ {0} → Y such that f (x) ∈ Zv(x) for all x ∈ M \ {0}. By Lemma 2.14, there is some δ ∈ ΓM such T that f is continuous on B(0, δ) \ {0}. By (4), f has a cluster point a ∈ Y . Then a ∈ γ Zγ . Otherwise, take γ large enough that a ∈ / Zγ . Because a is a cluster 8 point and Zγ is closed in Y , there is some x 6= 0 such that v(x) ≥ γ and f (x) ∈ / Zγ . By choice of f , f (x) ∈ Zv(x) ⊆ Zγ , a contradiction. (5)⇒(1): We first claim that Y is closed. Take p ∈ Y . Because X is a definable manifold, we can identify a neighborhood of p in X with the closed ball R(M)n in M n . For γ ≥ 0, let Bγ be the closed ball of radius γ around p. For γ ≤ 0 let Bγ = B0 . Then Bγ ∩ YTis a non-empty closed subset of Y for any γ, because p ∈ Y . By (4), the intersection γ (Bγ ∩ Y ) is non-empty, and so p ∈ Y . Therefore Y is closed. Similarly, Y is bounded. Take a Γ-exhaustion {Uγ }γ∈ΓM of the definable manifold X. If Y is unbounded, then Y \ Uγ is a closed non-empty subset of Y for each γ. Applying S (5) to the family of sets Y \ Uγ , we see that Y 6⊆ γ Uγ = X, a contradiction. Therefore Y is closed and bounded. By Proposition 2.10(2), Y is definably compact. Therefore, we could alternatively define definable compactness as follows: Proposition 2.16. Let Y be a definable subset of a definable manifold X. Then Y is definably compact if and only if every definable continuous function f : R(M)\{0} → Y has a cluster point. This is essentially the definition of “definable compactness” appearing in [13] (with the mistake fixed). 2.3 Definable compactness and definable 1-dimensional types Suppose that N ≻ M. Let X be a definable manifold in M. Definition 2.17. For a ∈ X(M) and b ∈ X(N), say that a and b are infinitesimally close over M if b is contained in every M-definable neighborhood of a. Suppose that X, Y are M-definable manifolds and f : X → Y is an M-definable continuous function. If a ∈ X(M) is infinitesimally close to b ∈ X(N), then f (a) is infinitesimally close to f (b). Definition 2.18. • We let OX(M ) (N) denote the set of b ∈ X(N) such that b is infinitesimally close to at least one a ∈ X(M). • There is a function stN M : OX(M ) (N) → X(M) sending each b to the unique a ∈ X(M) such that b and a are infinitesimally close. This is well-defined because X is Hausdorff. The map stN M is the “standard part” map from OX(M ) (N) to X(M). Definition 2.19. If p is a complete X-type over M, we say that p specializes to a ∈ X(M) if p(x) ⊢ x ∈ U for every M-definable neighborhood U ∋ a. If b ∈ X(N) is a realization of p, then p specializes to a if and only if stN M (b) = a. 9 ′ Fact 2.20. If aT ∈ N\M is infinitesimally close to a ∈ M over M, then there is a coset C ⊆ N\{0} of n≥1 Pn (N) such that tp(a′ /M) is determined by the partial type {v(x − a) > γ | γ ∈ ΓM } ∪ {x − a ∈ C}, and tp(a′ /M) is definable over M. This follows by a similar argument to Lemma 2.1 in [14]. Lemma 2.21. Let C be a definable (i.e., interpretable) family of balls B ⊆ M. Suppose the following conditions hold: 1. C is non-empty. 2. C is a chain: it is linearly ordered by ⊆. 3. C is upwards-closed: if B ⊇ B ′ ∈ C for balls B, B ′ , then B ∈ C. 4. C has no minimal element. Then there is d ∈ M such that C is the set of balls containing d. Proof. We may assume M = Qp , in which case the lemma is an easy exercise using spherical completeness of Qp . Lemma 2.22. Let X be an M-definable set, and p be a 1-dimensional definable type over M in X. Then there is an elementary extension N ≻ M and elements a ∈ M, b ∈ X(M), such that a is infinitesimally close to 0, b ∈ dcl(Ma), and p = tp(b/M). Proof. Take N ≻ M containing a realization b of p. Because p is 1-dimensional, there is some singleton c ∈ N such that dcl(Mb) = dcl(Mc). (In fact, we can take c to be a coordinate of the tuple b.) Replacing c with 1/c if necessary, we may assume that v(c) ≥ 0. Then tp(c/M) is definable and one-dimensional. Let C be the family of M-definable balls which contain c. Then C is definable, because tp(c/M) is definable. Moreover, C satisfies the four conditions of Lemma 2.21: 1. C is non-empty, because it contains the ball R(M) of radius 0. 2. C is a chain, because any two balls which intersect are comparable, and C cannot contain two disjoint balls. 3. C is upwards-closed, trivially. 4. C has no least element. Otherwise, if B were the smallest M-definable ball containing c, then we could write B as a disjoint union of smaller balls B = B1 ∪ · · · ∪ Bp , and one of the Bi would belong to C. By Lemma 2.21, C is the class of balls around some point d. So there is some d ∈ M such that c is contained in every M-definable ball around d. Therefore, c is infinitesimally close to d over M. Take a = c − d. 10 Lemma 2.23. Let X be a definable manifold over M. Let Y be a definably compact definable subset of X. Let p be a definable 1-dimensional complete Y -type over M. Then p specializes to a point in Y . Proof. Let N be an ℵ1 -saturated elementary extension of M, and let M be a monster model extending N. Let pN be the heir of p over N. We first show that pN specializes to a point in Y (N). Take c ∈ Y (M) realizing pN . By Lemma 2.22, we can write c as g(a) for some N-definable function g : M → Y (M) and some a ∈ M infinitesimally close to 0 over N. Because N is ℵ1 -saturated, there is some u ∈ N such that a/u ∈ Pn (M) for all n. Replacing a with a/u, we may assume that a ∈ Pn (M) for all n. For each n, let Sn ⊆ Y (N) be the definable set of cluster points T of g ↾ Pn (N). Each Sn is closed, and non-empty by Lemma 2.15(2). The T intersection n Sn is filtered, and therefore nonempty by ℵ1 -saturation. Take b ∈ n Sn . Let Σ(x) be the partial type saying that x is infinitesimally close to 0, g(x) is infinitesimally close to b, and x ∈ Pn for all n. Then Σ(x) is finitely satisfiable, by choice of b. Take a′ ∈ M realizing Σ(x). By Fact 2.20, tp(a′ /N) = tp(a/N). Therefore a satisfies Σ(x), and so g(a) is infinitesimally close to b. It follows that pN (x) specializes to b. Let Z be the set of b ∈ Y (N) such that pN specializes to b. The set Z is M-definable, because pN is definable over M. The above argument shows |Z| > 0. On the other hand, |Z| ≤ 1 because Y (N) is Hausdorff. Therefore Z is a singleton {b}, and the element b lies in Y (M). Then p specializes to b. Proposition 2.24. Work in a model M. Let X be a definable manifold and Y be a definable subset. Then Y is definably compact if and only if every 1-dimensional definable Y -type specializes to a point of Y . Proof. One direction is Lemma 2.23. Conversely, suppose every 1-dimensional definable type in Y specializes to a point. We claim that Y is definably compact. We use criterion (3) of Lemma 2.15. Let f : R(M) \ {0} → Y be a definable continuous function. Take a monster model M ≻ M and a non-zero a ∈ M infinitesimally close to 0 over M. Let b = f (a). By Fact 2.20, tp(a/M) is definable. Therefore tp(b/M) is 1-dimensional and definable. Then tp(b/M) specializes to a point c ∈ Y (M). We claim that c is a cluster point of f . For any M-definable neighborhoods U1 ∋ 0 and U2 ∋ c, we have (a, f (a)) = (a, b) ∈ U1 × U2 . As M ≺ M, there must be some (a′ , f (a′ )) ∈ U1 (M) × U2 (M). This shows that c is a cluster point of f . Lemma 2.25. Let X be an M-definable manifold and {Ot }t∈ΓM be a Γ-exhaustion. Let p be a definable 1-dimensional type in X over M, such that p does not concentrate on Ot for any t ∈ ΓM . Suppose M ≻ N ≻ M. Suppose that b ∈ X(M) realizes p, and b∈ / Ot (M) for any t ∈ ΓN . Then b realizes pN , the heir of p over N. Proof. By Lemma 2.22, we have b = f (a) for some M-definable function f : M → X and some a ∈ M infinitesimally close to 0 over M. By Lemma 2.14, f is continuous on B(0, γ0 ) for some sufficiently large γ0 ∈ ΓM ; note that v(a) > γ0 . We claim that a is infinitesimally close to 0 over N. Otherwise, there is some γ ∈ ΓN such that v(a) < γ. Let A be the definable set of x ∈ M such that γ0 < v(x) < γ; note that a ∈ A. The set A is definably compact and N-definable. Also, f is N-definable and 11 continuous on A. Therefore, the image f (A) is N-definable, and definably compact. By Proposition 2.10, there is some t ∈ ΓN such that f (A) ⊆ Ot . Then b = f (a) ∈ f (A) ⊆ Ot (M), contradicting the assumptions. This shows that a is infinitesimally close to 0 over N. By Fact 2.20, tp(a/N) is the heir of tp(a/M), implying that tp(b/N) = tp(f (a)/N) is the heir of tp(f (a)/M) = p. 2.4 Definable groups in pCF By a definable group over M, we mean a definable set with a definable group operation. By [18], any group G definable in M admits a unique definable manifold structure making the group operations be continuous. Remark 2.26. In particular, there is a canonical notion of “definable compactness” for abstract definable groups and their definable subsets. As in Remarks 2.3 and 2.11, one can show that these notions are definable in families and invariant in elementary extensions. Definition 2.27. A good neighborhood basis is a definable neighborhood basis of the form {Ot : t ∈ ΓM } which is also a Γ-exhaustion, and such that Ot = Ot−1 for each t ∈ ΓM . Proposition 2.28. Every definable group has a good neighborhood basis. Proof. By Proposition 2.8, the group G admits a Γ-exhaustion {Wt : t ∈ ΓM }. Replacing {Wt } with {Wt+γ }, we may assume that W0 is non-empty. Replacing {Wt } with {a·Wt }, we may assume that idG ∈ W0 . Because G is a definable manifold, there is some definable neighborhood basis {Nt : t ∈ ΓM , t < 0} such that each Nt is clopen, and Nt depends monotonically on t. Define ( Wt t≥0 Bt = W0 ∩ Nt t < 0. Then {Bt : t ∈ ΓM } is a definable neighborhood basis and a Γ-exhaustion. Lastly, define Ot = Bt ∩ Bt−1 . Then {Ot : t ∈ ΓM } has all the desired properties. Proposition 2.29. Let {Ot : t ∈ ΓM } be a good neighborhood basis of a definable group G. 1. For any t ∈ ΓM , there is t′ ∈ ΓM such that Ot′ · Ot′ ⊆ Ot . 2. For any t ∈ ΓM , there is t′′ ∈ ΓM such that Ot · Ot ⊆ Ot′′ . Proof. (1) is by continuity. For (2), note that the set Ot · Ot is an image of the definably compact space Ot × Ot under the definable continuous map (x, y) 7→ x · y. Therefore Ot · Ot is definably compact. Then t′′ exists by Proposition 2.10. Lemma 2.30. Let {Ot : t ∈ ΓM } be a good neighborhood basis of a definable group G. For every t, ǫ ∈ ΓM , there is δ ∈ ΓM such that if a ∈ Oδ and b ∈ Ot , then b−1 ab ∈ Oǫ . 12 Proof. Define Sδ = {(a, b) ∈ Oδ ×Ot : b−1 ab ∈ / Oǫ }. Suppose for the sake of contradiction that Sδ 6= ∅ for all δ. The family Sδ is definable, and depends monotonically on δ. Each set Sδ is closed, because Oǫ , Oδ , and Ot are clopen. By definable compactness of Oδ ×Ot , T the intersection δ Sδ is non-empty. Therefore there are a, b ∈ G such that 1. a ∈ Oδ for all δ. 2. b ∈ Ot . 3. b−1 ab ∈ / Oǫ . The first point implies a = idG , which then implies b−1 ab = idG ∈ Oǫ , a contradiction. 3 Review of dp-rank In Section 4 we will make extensive use of dp-rank, so we review its basic properties here. Definition 3.1. Let κ be a cardinal and Σ(x) be a partial type. An ict-pattern of depth κ in Σ(x) consists of • A family of formulas {φα (x, yα )}α<κ . • An array of parameters {bα,i }α<κ, i<ω with |bα,i | = |yα|. such that for any function η : κ → ω, the following type is consistent: Σ(x) ∪ {φα (x, bα,i ) : α < κ, i = η(α)} ∪ {¬φα (x, bα,i ) : α < κ, i 6= η(α)}. Definition 3.2. The dp-rank of a partial type Σ(x) is the supremum of cardinals κ such that, in some elementary extension N  M, there is an ict-pattern of depth κ in Σ(x). When there is no supremum, the dp-rank is defined to be ∞, a formal symbol greater than all cardinals. We write the dp-rank of Σ(x) as dp-rk(Σ). When Σ(x) is a complete type tp(b/A), we write the dp-rank as dp-rk(b/A). The following facts can be found in [10], or alternatively [22, Chapter 4]. Fact 3.3. The following are equivalent in a structure M: 1. M is NIP. 2. dp-rk(x = x) < ∞. 3. Every partial type has dp-rank < ∞. Fact 3.4. If Σ(x) is a partial type over A, and if the ambient model M is |A|+ -saturated, then dp-rk(Σ) is the supremum of dp-rk(b/A) as b ranges over realizations of Σ(x). Fact 3.5. If b ∈ acl(A), then dp-rk(b/A) = 0. If b ∈ / acl(A), then dp-rk(b/A) > 0. 13 Fact 3.6. For any b, c, A, we have dp-rk(b/A) ≤ dp-rk(bc/A) ≤ dp-rk(b/cA) + dp-rk(c/A). It is also helpful to view dp-rank as a property of definable sets: Definition 3.7. If D is a definable set, then the dp-rank of D, written dp-rk(D), is dp-rk(φ(x)) for any formula φ(x) defining D. The following facts are easy exercises using Facts 3.3–3.6. Fact 3.8. dp-rk(D) > 0 if and only if D is infinite. Fact 3.9. If D1 , D2 are definable sets, then dp-rk(D1 × D2 ) = dp-rk(D1 ) + dp-rk(D2 ). Fact 3.10. If f : D1 → D2 is a definable injection, then dp-rk(D1 ) ≤ dp-rk(D2 ). If f : D1 → D2 is a definable surjection, then dp-rk(D1 ) ≥ dp-rk(D2 ). We will need the following about dp-rank in p-adically closed fields: Fact 3.11 ([3, Theorem 6.6]). If M is a p-adically closed field, then dp-rk(M) = 1. Corollary 3.12. If M is a p-adically closed field, then every n-type in M has dp-rank at most n. In fact, dp-rank in pCF agrees with the natural notion of dimension (topological dimension or acl-dimension), by [22, Exercise 4.38]. We will not need this fact, however. 4 Large gaps In order to apply the strategy of Peterzil and Steinhorn, we need a technical statement about “gaps” in unbounded curves: Conjecture 4.1. Let G be a definable group over a p-adically closed field M, with a good neighborhood basis {Ot | t ∈ ΓM }. Let I be a 1-dimensional unbounded definable subset of G. Then for every t0 ∈ ΓM , there is t ∈ ΓM such that {g ∈ I | g(Ot \ Ot0 ) ∩ I = ∅} is bounded. The o-minimal analogue of Conjecture 4.1 holds by an easy connectedness argument [16, Lemma 3.8]. But in a p-adically closed field, everything is totally disconnected and we need a completely different approach. In the end, we will prove Conjecture 4.1 only in a special case (Proposition 4.14), namely when G is nearly abelian (Definition 1.1). Remark 4.2. It is useful to consider what a counterexample to Conjecture 4.1 would look like. For each t ≫ t0 , there would be unboundedly many g ∈ I such that g(Ot \ Ot0 ) ∩ I = ∅, or equivalently gOt ∩ I = gOt0 ∩ I. Around g, the set I looks like an “island” gOt0 ∩ I surrounded by a very large empty space g(Ot \ Ot0 ). Since I is unbounded, there must be infinitely many of these “islands.” Because this holds for any t, the gaps between the islands must become greater and greater as we move towards “∞”. 14 The behavior described above is reminiscent of the behavior of the set 2Z in the group (R, +, <). The structure (R, +, <, 2Z) is NIP [5, Theorem 6.5] but it does not have finite dp-rank, and this is a direct consequence of the “large gaps” in 2Z . In a non-standard elementary extension, by choosing a1 < b1 < a2 < b2 < · · · < an < bn carefully, one can ensure that the map n Y (2Z ∩ [ai , bi ]) → R i=1 (x1 , . . . , xn ) 7→ n X xi i=1 is injective and each set 2Z ∩ [ai , bi ] is infinite, showing that the model has dp-rank at least n (for arbitrary finite n). Our approach for attacking Conjecture 4.1 is based on this line of argument: take a set I with large gaps and obtain infinite dp-rank. Unfortunately, the argument only works in the nearly abelian case (Proposition 4.14), though we can salvage a much weaker statement in the non-abelian case (Proposition 4.15). 4.1 Notation Let G be a group. If H is a subgroup of G, we let G/H denote the set of left cosets of H. If A ⊆ G, we will write A/H to indicate the image of A in G/H. If A, B ⊆ G, we let AB indicate {b−1 ab : a ∈ A, b ∈ B}. Notation like “X \ Y ” will always mean set subtraction, rather than quotienting by a group action on the left. Definition 4.3. Let X, Y be subsets of a group G. Define X ⋄ Y = {g ∈ X : gY ∩ X = ∅}. Note that X ⋄ Y depends negatively on Y . We will write “A ⋄ B \ C” to mean “A ⋄ (B \ C).” Remark 4.4. Suppose X, Y are subgroups of G, S ⊆ G, and a, b ∈ S ⋄ X \ Y . Then aX = bX =⇒ aY = bY. Otherwise, b = aδ for some δ ∈ X \ Y , and so b ∈ a(X \ Y ) ∩ S, contradicting the fact that a(X \ Y ) ∩ S = ∅. 4.2 The bad gap configuration Recall that an externally definable set X in a structure M is a set of the form Y ∩ M n for some elementary extension N ≻ M and definable set Y ⊆ N n . The Shelah expansion M Sh is the expansion of M by all externally definable sets. When M is NIP, the Shelah expansion M Sh has elimination of quantifiers [22, Proposition 3.23]. Using this, it is easy to see that M Sh has the same dp-rank as M. 15 Remark 4.5. Let F be a collection of definable subsets of M n . If the S sets in FTare uniformly definable, and F is linearly ordered by inclusion, then the sets F and F are externally definable [6, Kaplan’s Lemma 3.4]. Later, we will use Remark 4.5 in conjunction with Proposition 2.29 to construct externally definable subgroups of definable groups. Definition 4.6. Let G be a definable group in a structure M. A bad gap configuration in G consists of the following • A finite subgroup F ⊆ G. • Externally definable subgroups · · · ⊆ Y 2 ⊆ Y 1 ⊆ Y 0 ⊆ X0 ⊆ X1 ⊆ · · · ⊆ G • An externally definable subset I ⊆ G. such that the following conditions hold: • YiF ⊆ Yi for all i. • YiXi ⊆ Yi−1 , for i > 0. • (Xi ∩ (I ⋄ Xi−1 \ F Yi−1 ))/Xi−1 is infinite, for i > 0. We say that a bad gap configuration is (A-)definable if all of F , the Xi , Yi , and I are (A-)definable. Lemma 4.7. If G has finite dp-rank, then there is no bad gap configuration in G. Proof. Let (F, {Yi }, {Xi }, I) be a bad gap configuration. Replacing M with the Shelah expansion M Sh , we may assume that the bad gap configuration is definable. Passing to an elementary extension and naming parameters, we may assume that M is ℵ1 -saturated and the bad gap configuration is ∅-definable. Note that F Yi = Yi F is a subgroup of G, and that the index of Yi if F Yi is finite, no more than |F |. Let Di be the definable set Xi ∩ (I ⋄ Xi−1 \ F Yi−1 ). By assumption, Di /Xi−1 is infinite. Claim 1. Suppose ai , a′i ∈ Di for i = 1, . . . , n, and suppose Yn an an−1 · · · a1 = Yn a′n a′n−1 · · · a′1 . (1) Then an F Yn−1 = a′n F Yn−1 . If moreover an Yn−1 = a′n Yn−1 , then Yn−1 an−1 · · · a1 = Yn−1 a′n−1 · · · a′1 . 16 (2) Proof. Note that ai , a′i ∈ Xi . Equation (1) implies that a′n a′n−1 · · · a′1 = ǫan an−1 · · · a1 = an ǫan an−1 · · · a1 (3) for some ǫ ∈ Yn . Then ǫan ∈ YnXn ⊆ Yn−1 ⊆ Xn−1 . For i < n, we have ai , a′i ∈ Xi ⊆ Xn−1 . Therefore (3) implies that a′n Xn−1 = an Xn−1 . Both a′n and an are in I ⋄ Xn−1 \ F Yn−1, so by Remark 4.4 we have a′n F Yn−1 = an F Yn−1 as desired. Now suppose that an Yn−1 = a′n Yn−1 . Then a′n = an δ for some δ ∈ Yn−1. Then Equation (3) implies an ǫan an−1 · · · a1 = an δa′n−1 a′n−2 · · · a′1 ǫan an−1 · · · a1 = δa′n−1 a′n−2 · · · a′1 . Both ǫan and δ are in Yn−1 , so Equation (2) holds. Claim For any n ∈ N, we claim that dp-rk(G) ≥ n. By assumption, the interpretable set Di /Xi−1 is infinite. The interpretable set DQ i /Yi−1 is even bigger, because Yi−1 ⊆ Xi−1 . n By the properties Qn of dp-rank in Section 3, i=1 Di /Yi−1 has dp-rank at least n. Take a tuple b̄ ∈ i=1 Di /Yi−1 such that dp-rk(b̄/∅) ≥ n. Each bi is a coset ai Yi−1 for some ai ∈ Di . Let c = an an−1 · · · a1 ∈ G. Claim 2. For each i, we have bi ∈ acl(c, bi+1 , . . . , bn ). Q Proof. Let S be the set of (a′1 , . . . , a′n ) ∈ j Dj such that • a′n a′n−1 · · · a′1 = c. • a′j Yj−1 = bj = aj Yj−1 for j > i. Then (a1 , . . . , an ) ∈ S and S is definable over c, bi+1 , . . . , bn . If (a′1 , . . . , a′n ) ∈ S, then Yn a′n a′n−1 · · · a′1 = Yn c = Yn an an−1 · · · a1 . By Claim 1 applied (n − i + 1) times, we see that ai F Yi−1 = a′i F Yi−1. We have shown {a′i F Yi−1 : (a′1 , . . . , a′n ) ∈ S} = {ai F Yi−1 }. It follows that ai F Yi−1 is definable over c, bi+1 , . . . , bn . The fibers of the map G/Yi−1 → G/(F Yi−1) are finite, and so ai Yi−1 = bi is algebraic over c, bi+1 , . . . , bn . Claim By Claim 2 and induction, b̄ ∈ acl(c). Therefore n ≤ dp-rk(b̄/∅) ≤ dp-rk(c/∅) ≤ dp-rk(G). As n was arbitrary, G has infinite dp-rank, a contradiction. 17 4.3 The saturated case Until Subsection 4.4, we will work in a monster model M |= pCF. Fix a definable group G, not definably compact, and fix a good neighborhood basis {Ot : t ∈ ΓM } in the sense of Definition 2.27. Lemma 4.8. There is no bad gap configuration in G. Proof. For definable sets in pCF, dp-rank agrees with dimension. In particular, dp-rank is finite. Therefore Lemma 4.7 applies to G. Recall from Definition 2.9 and Proposition 2.10(1) that a subset S ⊆ G(M) is bounded if S ⊆ Ot for some t ∈ ΓM . Lemma 4.9. If S ⊆ G(M) is bounded, then S ⊆ X for some bounded externally definable subgroup X ⊆ G(M). Proof. Take t0 ∈ ΓM such that S ⊆ Ot0 . By Proposition 2.29, we can build an ascending sequence t0 < t1 < t2 < · · · in ΓM such that Oti · Oti S ⊆ Oti+1 for each i. By saturation, we can also find some tω > ti for all finite i. Set X = i<ω Oti . The set X is externally definable (Remark 4.5). The for each i. set X is bounded, because X ⊆ Otω . We have X = X −1 because Oti = Ot−1 i Lastly, X is closed under the group operation by choice of the ti ’s. Lemma 4.10. Let I be an unbounded subset of G. Let X ⊆ G(M) be a bounded subgroup. Then there is an externally definable bounded subgroup X ′ ⊇ X such that (X ′ ∩ I)/X is infinite. Proof. We claim Sn that I/X is infinite. Otherwise, I is contained in a finite union of cosets: I ⊆ i=1 ai X.STake t ∈ ΓM such that X ⊆ Ot . Then I is a subset of the definably compact set ni=1 ai Ot , so I is bounded, a contradiction. Now take a1 , a2 , a3 , . . . ∈ I such that the cosets ai X are pairwise distinct. By saturation, there is some t ∈ Γ such that {a1 , a2 , . . .} ⊆ Ot . Then {a1 , a2 , . . .} and X are bounded. By Lemma 4.9, there is an externally definable bounded subgroup X ′ containing {a1 , a2 , . . .} ∪ X. Then (X ′ ∩ I)/X is infinite, witnessed by the ai X. Recall from Definition 1.1 that G is nearly abelian if there is a definably compact definable normal subgroup K ⊆ G with G/K abelian. Equivalently, G is nearly abelian if there is a definably compact subgroup K containing the derived group [G, G]. Lemma 4.11. Suppose that G is nearly abelian. Let I be an unbounded definable subset of G(M). For any bounded set A, there is t ∈ ΓM such that I ⋄ Ot \ A is bounded. Proof. Suppose not. Claim. For any bounded sets C ⊇ B ⊇ A, the set I ⋄ C \ B is unbounded. Proof. Take t ∈ ΓM such that C ⊆ Ot . Then I ⋄ C \ B contains the unbounded set I ⋄ Ot \ A, because Ot \ A ⊇ C \ B. Claim 18 Let K be the normal subgroup witnessing near-abelianity. By Lemma 4.9, there is a bounded externally definable subgroup X0 ⊇ A ∪ K. By Lemma 4.10 we can recursively build an increasing chain of bounded externally definable subgroups X0 ⊆ X1 ⊆ X2 ⊆ · · · such that • (X1 ∩ I)/X0 is infinite. • For n > 1, (Xn ∩ (I ⋄ Xn−1 \ X0 ))/Xn−1 is infinite. This is possible because I ⋄ Xn−1 \ X0 is unbounded by the claim. Let Yi = X0 for all i, and let F = {idG }. Note X0 is normal, because it contains K which contains [G, G]. We have constructed a bad gap configuration in G, contradicting Lemma 4.8. Lemma 4.12. If S ⊆ G(M) is a neighborhood of idG , then S ⊇ X for some externally definable open subgroup X ⊆ G(M). If, in addition, B ⊆ G(M) is a bounded set, then we can choose the group X to ensure X B ⊆ X. Proof. Take t0 ∈ ΓM such that S ⊇ Ot0 . By Proposition 2.29 and Lemma 2.30, there is a descending sequence t0 > t1 > t2 > · · · T in ΓM such that Oti+1 · Oti+1 ⊆ Oti and also OtBi+1 ⊆ Oti . Take X = ∞ i=1 Oti . Then X is B an externally definable subgroup with X ⊆ X. We can take some tω less than all the ti ’s, and then Otω ⊆ X. Therefore X has interior, and is an open subgroup. Lemma 4.13. Let I be an unbounded definable subset of G(M). Let F be a finite subgroup of G(M). Then there exist t, t′ ∈ ΓM such that I ⋄ Ot \ (F · Ot′ ) is bounded. Proof. Suppose not. Claim. For any neighborhood A ∋ idG and any bounded set B ⊆ G, the set I ⋄ B \ F A is unbounded. Proof. Take t, t′ such that Ot′ ⊆ A and B ⊆ Ot Ot \ (F · Ot′ ) ⊇ B \ (F · A) I ⋄ Ot \ (F · Ot′ ) ⊆ I ⋄ B \ (F · A). Claim Take any bounded open externally definable subgroup X0 ⊆ G. By Lemma 4.12 there is an externally definable open subgroup Y0 ⊆ X0 such that Y0F ⊆ Y0 . Recursively build chains X0 ⊆ X1 ⊆ · · · Y0 ⊇ Y1 ⊇ · · · where 19 • Xi is a bounded externally definable subgroup, chosen large enough to ensure that (Xi ∩ (I ⋄ Xi−1 \ F Yi−1 ))/Xi−1 is infinite (Lemma 4.10). • Yi is an open externally definable subgroup with YiF = Yi , chosen small enough that YiXi ⊆ Yi−1 (Lemma 4.12). This gives a bad gap configuration in G, contradicting Lemma 4.8. 4.4 The general case Proposition 4.14. Let M be any model of pCF. Let G be a definable non-compact group and {Ot : t ∈ ΓM } be a good neighborhood basis. Suppose that G is nearly abelian. Let I be an unbounded definable set. Then for any t ∈ ΓM , there is t′ ∈ ΓM such that I ⋄ Ot′ \ Ot is bounded. Proof. We may replace M with a monster model, and then apply Lemma 4.11. Proposition 4.15. Let M be any model of pCF. Let G be a definable non-compact group. Let I be an unbounded definable set. Let F be a finite subgroup of G. Then for any sufficiently small s and sufficiently large t, the set I ⋄ Ot \ (F Os ) is bounded. Proof. We may replace M with a monster model, and then apply Lemma 4.13. 5 Stabilizers and µ-stabilizers In this section we review some notation and facts from [15]. 5.1 Stabilizers Let G be a group definable in a structure M. Notation 5.1. (1) If φ(x) and ψ(x) are G-formulas then φ · ψ denotes the G-formula (φ · ψ)(x) := ∃u∃v(φ(u) ∧ ψ(v) ∧ x = u · v). Thus (φ · ψ)(M) = φ(M) · ψ(M). (2) More generally, if q(x) and r(x) are partial G-types then q · r denotes the G-type (q · r)(x) := {φ · ψ(x) | q(x) ⊢ φ(x), r(x) ⊢ ψ(x)}. Thus (q·r)(N) = q(N)·r(N) for an |M|+ -saturated elementary extension N ≻ M. (3) If g ∈ G(M) and φ(x) is a G-formula, then g · φ denotes the G-formula (g · φ)(x) := ∃u(φ(u) ∧ x = g · u). Thus (g · φ)(M) = g · φ(M). 20 (4) If g ∈ G(M) and p(x) is a partial G-type then g · p denotes the G-type (g · p)(x) := {g · φ(x) | p(x) ⊢ φ(x)}. Thus (g · p)(N) = g · p(N) for an |M|+ -saturated N ≻ M. Note that for partial G-types q1 , q2 , q3 over M, we have (q1 · q2 ) · q3 = q1 · (q2 · q3 ), as ((q1 · q2 ) · q3 )(N) = q1 (N) · q2 (N) · q3 (N) = (q1 · (q2 · q3 ))(N) for |M|+ -saturated N ≻ M. Definition 5.2. Given a partial type Σ(x) over M, define stab(Σ) to be the stabilizer, i.e., stab(Σ) := {g ∈ G(M) | gΣ ≡ Σ}, where Σ ≡ Σ′ if Σ(x) ⊢ Σ′ (x) and Σ′ (x) ⊢ Σ(x). Equivalently, stab(Σ) is {g ∈ G(M) | gΣ(N) = Σ(N)} for |M|+ -saturated N  M. Definition 5.3. Given a partial type Σ(x) over M and an L-formula φ(x, y), we define \ stabφ (Σ) = Xφ,b , b∈M k where each Xφ,b is the stabilizer of {g ∈ G(M) | Σ ⊢ (gφ)(x, b)}. Remark 5.4. Given φ(x; y), let φ′ (x; y, z) be the formula φ(z · x; y). Then G acts on φ′ -types by left translation, and stabφ (Σ) is the stabilizer of the φ′ -type generated by Σ. Remark 5.5. Note that our stabφ is slightly different from the Stabφ considered in [15], which is more like the set Xφ,b appearing in Definition 5.3 above. The following two facts are easy exercises. Fact 5.6. stabφ (Σ) is a definable subgroup of G if Σ is definable. Fact 5.7. For every partial type Σ over M. \ stab(Σ) = stabφ (Σ) φ∈L In particular, if Σ is definable then stab(Σ) is an intersection of definable subgroups. Recall the notation ΣN for the canonical extension of a definable type Σ to an elementary extension N ≻ M, and the notation (dΣ x)φ(x; y) for the φ-definition of Σ. Lemma 5.8. If Σ is definable and N ≻ M, then stabφ (ΣN ) = stabφ (Σ)(N), and so \ stabφ (Σ)(N). stab(ΣN ) = φ∈L Proof. Indeed, stabφ (Σ)(M) is defined by the formula ∀y∀g : ((dΣ z)φ(g · z; y)) ↔ ((dΣ z)φ(x · g · z; y)) and stabφ (ΣN ) is defined by the same formula, because ΣN and Σ have the same definition schema. 21 5.2 µ-types and µ-stabilizers In this section we assume that G is a Hausdorff topological group definable in M with a uniformly definable basis {Ot | t ∈ T } of open neighborhoods of the identity. For each N ≻ M, the group G(N) is again a topological group and the definable family {Ot (N) | t ∈ T (N)} again forms a basis for the open neighborhoods of idG . Definition 5.9. The infinitesimal type of G, denoted µ(x), is the partial type consisting of all formulas x ∈ U with U an M-definable neighborhood of idG . Thus, if N  M, then µ(N) is the set of elements of G(N) which are infinitesimally close to idG : \ µ(N) = {U(N) | U is an M-definable neighborhood of idG } \ = Ot (N). t∈T (M ) Fact 5.10 ([15, Corollary 2.5 and Claim 2.15]). 1. If N ≻ M, then µ(N) is a subgroup of G(N) normalized by G(M). 2. For any definable q ∈ SG (M), the partial type µ · q is definable. Partial types of the form µ · q for q ∈ SG (M) are called µ-types. The µ-stabilizer of q ∈ SG (M) is the stabilizer of the associated µ-type: stabµ (q) := stab(µ · q). Note that if µ is the infinitesimal type of G = G(M), and N  M, then the canonical extension µN is the infinitesimal type of G(N). Fact 5.11 ([15, Remark 2.16]). If p is a definable type over M and N ≻ M, then the product of the canonical extensions is equal to the canonical extension of the product: µN · pN = (µ · p)N . Remark 5.12. Let N be an |M|+ -saturated extension of M, and µN and pN be the canonical extensions of µ and p. Then \ stab(µN · pN ) = stab((µ · p)N ) = stabφ (µ · p)(N), φ∈L by Lemma 5.8 and Fact 5.11. By Fact 5.10, µ(N) · G(M) is a subgroup of G(N) as µ(N) ⊆ G(N) is normalized by G(M). This subgroup is the OG(M ) (N) of Definition 2.18. Because µ(N) ∩ G(M) = {idG }, the group µ(N) · G(M) is a semidirect product of µ(N) and G(M), and there is a natural homomorphism OG(M ) (N) = µ(N) · G(M) → G(M). This map is exactly the “standard part” map stN M of Definition 2.18. For Y ⊆ G(N), N we will write stN (Y ) as a shorthand for st (Y ∩ OG(M ) (N)), following [15]. M M 22 Lemma 5.13. Let p ∈ SG (M) be a definable type and let β ∈ G(M) realize pN . Then N −1 1. stab(µN · pN ) = stM ); N (p (M)β T −1 2. stab(µN · pN ) = ψ∈pN stM ); N (ψ(M)β Proof. Clause (1) is by Claim 2.22 in [15]. For (2), we must show N −1 stM )= N (p (M)β \ −1 stM ). N (ψ(M)β ψ∈pN T −1 The ⊆ direction is clear. For ⊇, suppose that g ∈ ψ∈pN stM ). Then for any N (ψ(M)β N −1 −1 N ψ ∈ p , there is hψ ∈ ψ(M) such that hψ · β · g satisfies µ . By compactness there N −1 is h ∈ pN (M) such that h · β −1 · g −1 satisfies µN . Then g ∈ stM ). N (p (M)β 6 Proof of main theorems From now on M is a p-adically closed field, M ≻ M is the monster model, G ⊆ M n denotes a group definable in M, and µ denotes the infinitesimal type of G over M. All formulas and types will be G-formulas and G-types. We assume G is not definably compact. Fix a good neighborhood basis {Ot : t ∈ ΓM } of G. Fix a 1-dimensional definable type p ∈ SG (M) which does not specialize to any point of G(M). Such a type p exists by Proposition 2.24. Fix a small |M|+ -saturated model N with M ≺ N ≺ M. As usual, pN and µN denote the canonical extensions to N. Fix an element β ∈ G(M) realizing pN . Remark 6.1. The types p and pN are “unbounded” in the following sense: 1. If t ∈ ΓM , then Ot ∈ / p. 2. If t ∈ ΓN , then Ot ∈ / pN . 3. If X is a bounded M-definable subset of G(M), then X ∈ / p. 4. If X is a bounded N-definable subset of G(N), then X ∈ / pN . Point (1) follows by Proposition 2.24: if Ot ∈ p then p specializes to a point in Ot (M), because Ot is definably compact. Point (2) then follows because pN is the heir of p. Points (3) and (4) reduce to (1) and (2), respectively. T −1 Lemma 6.2. stab(µN · pN ) = φ∈p stM ). N (φ(N)β Proof. By Lemma 5.13, it suffices to show \ \ −1 −1 (φ(M)β ) ⊆ stM ) stM N N (φ(M)β φ∈p φ∈pN Suppose g belongs to the left-hand side. In particular, g ∈ G(N). By a compactness argument similar to Lemma 5.13, we see that g = ǫbβ −1 for some ǫ ∈ µN (M) and 23 b ∈ p(M). It suffices to show b ∈ pN (M). By Lemma 2.25, it suffices to show b ∈ / Ot (M) ′ for any t ∈ ΓN . Suppose b ∈ Ot (M). Since g ∈ N, there is some t ∈ ΓN such that g −1 · O0 (M) · Ot (M) ⊆ Ot′ (M). Then β = g −1 ǫb ∈ g −1 O0 (M)Ot (M) ⊆ Ot′ (M), contradicting the fact that tp(β/N) is unbounded. Note that a similar argument to the proof of Lemma 2.31 of [23] shows the following: Fact 6.3. Suppose that b ∈ Mk and tp(b/N) is definable. If Y ⊆ G(M) is definable over b then stM N (Y ) ⊆ G(N) is definable and dim(stM N (Y )) ≤ dim(Y ). Lemma 6.4. There is an L-formula φ such that dim(stabφ (µ · p)) ≤ 1. −1 Proof. Take ψ ∈ p such that dim(ψ(M)) = 1. Then dim(stM N (ψ(M)β )) ≤ 1 by Fact 6.3. By Remark 5.12 and Lemma 5.13, \ −1 stabφ (µ · p)(N) = stab(µN · pN ) ⊆ stM ). N (ψ(M)β φ∈L The intersection on the left is directed, and the set on the right is definable, so by |M|+ -saturation of N there is some φ ∈ L such that −1 stabφ (µ · p)(N) ⊆ stM ). N (ψ(M)β −1 Then dim(stabφ (µ · p)) ≤ dim(stM )) ≤ 1. N (ψ(M)β To finish our main result, we now show that each stabφ (µ · p) is not definably compact. Lemma 6.5. Assume G is nearly abelian (Definition 1.1). For any N-definable set I −1 containing β, the set stM N (I(M)β ) is unbounded. −1 −1 Proof. Suppose stM ) is bounded. Then stM ) ⊆ Ot (N) for some t ∈ N (I(M)β N (I(M)β ΓN . By Remark 6.1, I is unbounded. By Proposition 4.14, there is some t′ ∈ ΓN such that the set I −1 ⋄ Ot′ \ Ot is bounded. Then β −1 ∈ / (I −1 ⋄ Ot′ \ Ot )(M) by Remark 6.1. This means that β −1 (Ot′ (M) \ Ot (M)) ∩ I(M)−1 6= ∅. Therefore there is a ∈ Ot′ (M) \ Ot (M) such that aβ ∈ I(M). By definable Skolem functions, we can take a ∈ dcl(Nβ). Note a ∈ I(M)β −1 . By Lemma 2.23, stM N (a) exists. Because Ot′ \ Ot is M closed, we see that stN (a) ∈ Ot′ (N) \ Ot (N). This contradicts the fact that M −1 stM ) ⊆ Ot (N). N (a) ∈ stN (I(M)β Lemma 6.6. If G is nearly abelian, then the type-definable group stab(µN · pN ) ⊆ G(N) is 1-dimensional and unbounded. 24 Proof. The dimension of stab(µN · pN ) is at most one by Lemma 6.4 and Remark 5.12. If stab(µN · pN ) is bounded, then stab(µN · pN ) ⊆ Ot (N) for some t ∈ ΓN . By Lemma 6.2, we have \ −1 stM ) = stab(µN · pN ) ⊆ Ot (N). N (ψ(N)β ψ∈p The intersection on the left is a filtered intersection of definable sets. There are at most |M| sets in the intersection, and N is |M|+ -saturated. Therefore there is some ψ ∈ p −1 such that stM ) ⊆ Ot (N), contradicting Lemma 6.5. Therefore stab(µN · pN ) N (ψ(N)β is unbounded. In particular, it is infinite, so it has dimension at least 1. Lemma 6.7. Suppose G is nearly abelian. Then there is φ ∈ L such that the M-definable group stabφ (µ · p) is not definably compact and has dimension 1. Proof. By Lemma 6.4 there is an L-formula φ such that dim(stabφ (µ · p)) ≤ 1. By Remark 5.12, we have stabφ (µ · p)(N) ⊇ stab(µN · pN ), and therefore stabφ (µ · p) is unbounded by Lemma 6.6. In particular, stabφ (µ · p) is infinite, and dim(stabφ (µ · p)) ≥ 1. Theorem 6.8. Let G be a definable group in a p-adically closed field M. Suppose G is nearly abelian, and not definably compact. 1. G has a one-dimensional definable subgroup which is not definably compact. 2. If p ∈ SG (M) is a definable unbounded 1-dimensional type, then there is φ ∈ L such that stabφ (µ · p) is a one-dimensional definable subgroup of G which is not definably compact. 3. Suppose in addition that M is ℵ1 -saturated. If p ∈ SG (M) is a definable unbounded 1-dimensional type, then stabµ (p) is a one-dimensional type-definable subgroup of G which is unbounded. Proof. Part (2) is Lemma 6.7. Part (1) then follows because there is at least one unbounded 1-dimensional definable type by Proposition 2.24. For Part (3), take a countable model M0  M such that G and p are M0 -definable. Then apply Lemma 6.6 to M and M0 in place of N and M (respectively), to see that stab(µ · p) is 1-dimensional and unbounded. Type-definability is by Fact 5.7. Recall from [19] that in an NIP context, a global type p ∈ SG (M) is said to be a definable f -generic, abbreviated as dfg, if there is a small submodel M0 such that every left G-translate of p is definable over M0 . In [19], Pillay and the second author showed that: Fact 6.9. A group G definable over Qp has dfg iff there is a normal sequence of definable subgroups G0 ⊳ ...Gi ⊳ Gi+1 ... ⊳ Gn such that G0 is finite, Gn is a finite index subgroup of G, and each Gi+1 /Gi is definably isomorphic to either the additive group Ga , or a finite index subgroup of the multiplicative group Gm . 25 The intuition is that “dfg” means “totally non-compact” in the p-adic context. Lemma 6.10. Let G be a one-dimensional definable group. If G is not definably compact, then G has dfg. Proof. Recall from [8, Section 4] that G has finitely satisfiable generics (fsg) if there is a small model M0 and a global type p(x) in G such that every left G-translate of p is finitely satisfiable in M0 . By [19, Lemma 2.9], G has fsg or dfg.1 It suffices to show that G does not have fsg. Suppose otherwise, witnessed by p and M0 . Recall that a definable set X ⊆ G is generic if finitely many left translates cover G. By [8, Proposition 4.2], the complement of a non-generic set is generic, and every generic set intersects G(M0 ). Let {Wγ }γ be a Γ-exhaustion of G. By taking γ sufficiently large, we can arrange for G(M0 ) ⊆ Wγ , because M0 is small. Then G \ Wγ does not intersect G(M0 ), so it is not generic. Therefore the complement Wγ is generic. A finite union of left translates of Wγ will be definably compact, so it cannot be all of G. We conclude that G does not have fsg, and instead has dfg. The following is then a corollary of Theorem 6.8. Corollary 6.11. Let G ⊆ M k be a nearly abelian definable group. If G is not definably compact, then G has a one-dimensional dfg subgroup. Next, we consider the general non-abelian case. Let G, M, N, M, p, β be as in the start of this section. Lemma 6.12. For any N-definable set I containing β, and any finite N-definable −1 subgroup F ⊆ G, the set stM N (I(M)β ) contains a point outside of F . Proof. As in Lemma 6.5, I is unbounded. Let J be the unbounded set I −1 . By Proposition 4.15, there are s, t ∈ ΓN such that J ⋄Ot \F Os is bounded. Then β −1 ∈ / J ⋄Ot \F Os. Therefore β −1 (Ot \ F Os ) ∩ J 6= ∅, or equivalently (Ot \ Os F )β ∩ I 6= ∅. Therefore there is a ∈ Ot (M) \ Os (M)F (M) such that aβ ∈ I(M). By definable Skolem functions, we can take a ∈ dcl(Nβ). Note a ∈ I(M)β −1 . Because Ot \ Os F is definably M compact, Lemma 2.23 implies that stM N (a) exists and is in Ot \ Os F . Then stN (a) is not in F , since F ⊆ Os F . Lemma 6.13. The group stab(µN · pN ) is 1-dimensional. In particular, it is infinite. Proof. As in Lemma 6.6, the dimension is at most 1. If dim(stab(µN · pN )) = 0, then stab(µN · pN ) is a finite subgroup F . By Lemma 6.2, we have \ −1 stM ) = stab(µN · pN ) = F. N (φ(N)β φ∈p −1 As in Lemma 6.6, there is some φ ∈ p such that stM ) ⊆ F . This contradicts N (φ(N)β Lemma 6.12 1 In [19] this is stated only for groups definable over Qp . The assumption is used in order to apply [20, Theorem 2.4]. However, [20, Remark 2.5] shows that this assumption is unnecessary. 26 By Lemma 6.4 and Remark 5.12, we conclude Lemma 6.14. There is φ ∈ L such that the M-definable group stabφ (µ · p) has dimension 1. We summarize the non-abelian case in the following theorem. Theorem 6.15. Let G be a definable group in a p-adically closed field M. Suppose G is not definably compact. Let p ∈ SG (M) be a definable unbounded 1-dimensional type. 1. There is φ ∈ L such that stabφ (µ · p) has dimension 1. 2. If M is ℵ1 -saturated, then stabµ (p) is a one-dimensional type-definable subgroup of G. Proof. Part (1) is Lemma 6.14. For Part (2), take a countable submodel M0  M such that G and p are M0 -definable. Then apply Lemma 6.13 with M and M0 in place of N and M. 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