Some model theory and topological dynamics of $p$-adic algebraic groups

arXiv:1704.07764v3 [math.LO] 18 Feb 2019 Some model theory and topological dynamics of p-adic algebraic groups Davide Penazzi ∗ University of Central Lancashire Anand Pillay † University of Notre Dame Ningyuan Yao‡ Fudan University February 19, 2019 Abstract We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Qp in the language of fields. We consider the additive and multiplicative groups of Qp and Zp , the group of upper triangular invertible 2 × 2 matrices, SL(2, Zp ), and, our main focus, SL(2, Qp ). In all cases we identify f -generic types (when they exist), minimal subflows, and idempotents. Among the main results is that the “Ellis group” of SL(2, Qp ) is Ẑ, yielding a counterexample to Newelski’s conjecture with new features: G = G00 = G000 but the Ellis group is infinite. A final section deals with the action of SL(2, Qp ) on the type-space of the projective line over Qp . 1 Introduction and preliminaries The machinery of topological dynamics has proved to be useful in generalizing stable group theory to unstable environments (the original paper on the topic being [19]). Given a structure M and group G definable in M, a natural action, given by model theory, is that of G on the space SG (M) of complete types over M concentrating on G. On the other hand this action is simply a dynamical system for G considered as a discrete group. When T h(M) is stable, G is what is called a stable group, and the fundamental theorems of stable group theory are coded in this dynamical system. There has been a ∗ Partially supported by IMA Small Grant SGS21/16 and NSF grant DMS-141947 Partially supported by NSF grants DMS-1360702, DMS-1665035, and DMS-1760413 ‡ Partially supported by NSFC grant 11601090, Initial Scientific Research Fund of Young Teachers in Fudan University, and William J. Hank Family Chair research funds (Notre Dame) † 1 considerable amount of work extending stable group theory to the case where T h(M) is NIP (does not have the independence property), and G is definably amenable (see [14] and [5] for example). When M is the field of reals, then a (semialgebraic) noncompact simple Lie group such as SL(2, R) is not definably amenable, but its definable topological dynamics was nevertheless analyzed in [10]. The latter work was partly motivated by a conjecture of Newelski on the connection between the Ellis group of such an action and the definable Bohr compactification G∗ /(G∗ )00 of G (G∗ being the interpretation of G in a saturated elementary extension). The case of SL( 2, R) gives an example where these two invariants are different, the definable Bohr compactification being trivial and the Ellis group being Z/2Z. In the current paper we extend this analysis of [10] to the p-adic context, namely where M is the field of p-adic numbers, rather than the field of reals. We focus on SL(2, Qp ) and its building blocks but the analysis should extend to semisimple p-adic Lie groups (as groups definable in the p-adic field). In the real case we made use of the Iwasawa decomposition of SL(2, R) as B(R)0 ·SO(2, R), where B is the Borel subgroup of upper triangular matrices, SO(2, R) is a maximal compact subgroup, and B(R)0 is the semialgebraic, equivalently topological, connected component of B(R) (note that the intersection of B(R)0 and SO(2, R) is trivial). In the p-adic case, the Iwasawa decomposition of SL(2, Qp ) has the form B(Qp ) · SL(2, Zp ) where B is as before, and now SL(2, Zp ) is a maximal compact subgroup. However now the intersection of the constituents is large (in fact of p-adic dimension 2) and also the constituents are far from connected. For example SL(2, Zp ), being profinite, has trivial connected component. So the analysis in the p-adic case is rather harder and requires some new ideas. A crucial role in our analysis of SL(2, R) was its action on the homogeneous space SL(2, R)/B(R)0 , which is a 2-cover of the natural action of SL(2, R) on P1 (R). In fact the universal minimal definable flow of SL(2, R) was the space of nonalgebraic types of the homogeneous space SL(2, R)/B(R)0. We proceed quite differently in the p-adic case. On the other hand, there are analogies between the final statements regarding the Ellis group; in the real case the Ellis group of SL(2, R) (acting on its type space) is Z/2Z which coincides with K∗ /(K∗ )0 where K∗ is the muultiplicative group of a saturated real closed field K. In the p-adic case, the Ellis of group of SL(2, Qp ) (acting on its type space) is Ẑ which coincides with K∗ /(K∗ )0 where K∗ is the multiplicative group a saturated p-adically closed field K. In any case, SL(2, Qp ) provides another counterexample to Newelski’s conjecture on the relationship between the Ellis group and G/G00 , but with different features from the ones provided by Corollary 0.3 of [16] for example, as the Ellis group is infinite whereas G = G00 = G000 . To be more precise, our main results are as follows where M denotes the structure (Qp , +, ×), G denotes SL(2, −), SG (M) denotes the space of complete types over M extending the formula ‘x ∈ G’, ∗ denotes the canonical semigroup structure on SG (M), and other notation will be explained later. • A minimal subflow of (G(M), SG (M)) is cl(I ∗ J ) where I is the unique minimal subflow of the action of SL(2, Zp ) on its type space, and J is a certain minimal 2 subflow of the action of B(Qp ) on its type space. In particular cl(I ∗ J ) is the universal minimal definable flow of SL(2, Qp ). See Theorem 3.4. • The Ellis group attached to the flow (G(M), SG (M)) is Ẑ. See Corollary 3.8. We also prove that the space of nonalgebraic types over M of the projective line P (Qp ) is minimal and proximal under the natural action of SL(2, Qp ). See Corollary 4.8. As part of our analysis we classify 1-types over Qp from the stable group theory point of view, namely we describe f -generics of various kinds (definable, finitely satisfiable) and minimal flows, with respect to the additive and multiplicative groups. This does not seem to have observed before, and provides interesting phenomena for definable topological dynamics in the NIP setting. Let us discuss where our work fits into current themes in topological dynamics and definable groups. This paper does not explicitly offer any new general results in topological dynamics and model theory. However, the project of generalizing the study of groups definable in o-minimal structures to the p-adic environment has been on the cards for a long time. Benjamin Druart’s thesis [7] and the preprint [8], studied groups definable in p-adically closed fields, in particular SL(2, Qp ), in analogy with o-minimal and finite Morley rank groups methods. On the other hand there has been considerable interest in generalising the stability-theoretic and topological dynamical study of real Lie groups such as SL(2, R) to the p-adic context since the paper [10] was written in 2012, and this is what we accomplish in the current paper. Moreover, as G. Jagiella has pointed out to us, the methods in our paper suggest generalizations to definable groups G in NIP theories with a decomposition G = B · K where B has “definable f -generics” and K has “finitely satisfiable generics”. This will be pursued in future work. Our notation for model theory is standard, and we will assume familiarity with basic notions such as type spaces, heirs, coheirs, definable types etc. References are [24] as well as [21]. Our notation for the p-adics is as follows: Qp is the field of p-adics and Zp is the ring of p-adic integers. Z is the ordered additive group of integers, the value group of Qp . M denotes the standard model (Qp , +, ×, −, 0, 1), and we sometimes write Qp for M. M̄ denotes a saturated elementary extension (K, +, ×, 0, 1) of M and again sometimes we write K for M̄ . Γ denotes the value group of K. We will be referring a lot to the comprehensive survey [2], for the basic model theory of the p-adics. A key point is Macintyre’s theorem [18] that T h(Qp , +, ×, 0, 1) has quantifier elimination in the language where we add predicates Pn (x) for the nth powers (all n). Moreover the valuation is quantifier-free definable in Macintyre’s language, in particular is definable in the language of rings. (See Section 3.2 of [2].) We will give a little more background at the beginning of Section 2.1. In the rest of this introduction we give more background on topological dynamics and the model-theoretic approach. In Section 2 we analyse the model-theoretic dynamics of the building blocks of SL(2, Qp ), namely the additive and multiplicative groups, the Borel subgroup, and the 1 3 maximal compact subgroup SL(2, Zp ). As mentioned earlier, this is of independent interest. In Section 3, we prove the main results, on the minimal subflows and Ellis group of the action on SL(2, Qp ) on its type space, making use of the Iwasawa decomposition and results in Section 2. In Section 4 we study the action on the type space of the projective line. We also ask several questions. We would like to thank a referee for his/her comments on a first version of this paper. He/she pointed out many mathematical points which needed clarification and/or correction, often suggesting the required correction. Following these comments, we have also added explanations of how the current paper differs from the earlier work ([10]) on SL(2, R), and how it relates to other current research. 1.1 Topological dynamics Our references for (abstract) topological dynamics are [1] and [12] Given a (Hausdorff) topological group G, by a G-flow mean a continuous action G × X → X of G on a compact (Hausdorff) topological space X. We sometimes write the flow as (X, G). Often it is assumed that there is a dense orbit, and sometimes a G-flow (X, G) with a distinguished point x ∈ X whose orbit is dense is called a G-ambit. In spite of p-adic algebraic groups being nondiscrete topological groups, we will be treating them as discrete groups so as to have their actions on type spaces being contiinuous. (But note that there is a model-theoretic account of the dynamics of definable groups with a definable topology. See [17] for example. And it might be worthwhile to prove and compare results. So in this background section we assume G to be a discrete group, in which case a G-flow is simply an action of G by homeomorphisms on a compact space X. By a subflow of (X, G) we mean a closed G-invariant subspace Y of X (together with the action of G on Y ). (X, G) will always have minimal nonempty subflows. Points x, y ∈ X are proximal with respect to (X, G) if there is a net (gα )α in G and z in X such that both gα x and gα y converge to z. (X, G) is proximal if every pair of elements of X is proximal. Given a flow (X, G), its enveloping semigroup E(X) is the closure in the space X X (with the product topology) of the set of maps πg : X → X, where πg (x) = gx, equipped with composition ◦ (which is continuous on the left). So any e ∈ E(X) is a map from X to X and, for example, proximality of the flow (X, G) is equivalent to: for all x, y ∈ X there is e ∈ E(X) such that e(x) = e(y). Note also that E(X) is a compact space and we have an action g · e = πg ◦ e of G on E(X), by homeomorphisms. So (E(X), G) is a flow too. Ellis [9] proved the following correspondence between minimal subflows and ideals of E(X). Theorem 1.1. Denote by J the set of idempotents of the enveloping semigroup E(X). Then 4 1. Minimal (automatically closed) left ideals I of E(X) coincide with minimal subflows. 2. Given a minimal closed left ideal I, I ∩ J 6= ∅; moreover for u ∈ I ∩ J, (u ◦ I, ◦) is a group, called the Ellis group. 3. All Ellis groups (varying I and u) are isomorphic, so we sometimes refer to this isomorphism class as the Ellis group attached to the original flow (X, G). There is a universal G-ambit, which is (under our discreteness assumption on G) the Stone-Cech compactification βG of G. This is precisely the (Stone) space of ultrafilters on the Boolean algebra of all subsets of G. The action of G on itself by left translation gives rise to an action on βG by homeomorphisms. Identifying g ∈ G with the principal ultrafilter it generates yields an embedding of G in βG and βG together with the identity element idG of G as distinguished point, is the universal G-ambit. The universal property is that for any other G-ambit (X, x) there is a unique map of G flows from βG to X which takes idG to X. The enveloping semigroup E(βG) of βG coincides with βG (due to its universal character) and hence βG is equipped with a canonical semigroup structure, continuous on the left. It can be described explicitly in various ways, see Section 4 of [19] for one such description. Minimal G-subflows, equivalently minimal left ideals, of βG are isomorphic as G-flows and coincide with the universal minimal G-flow (M, G), a G-flow, unique up to isomomorphism, with the feature that any minimal G-flow is an (surjective) image of (M, G) under a map of G-flows. The Ellis group of (βG, G) is an important invariant of the group G. 1.2 Model theory The model-theoretic background for the current paper is contained in [11] and [23], but we give a quick summary here. Given an L-structure M, S(M) denotes the collection of all complete types over M (in all sorts or number of variables). For a definable set Z in M, SZ (M) denotes the (Stone) space of complete types over M containing the formula x ∈ Z. If M ′ is an elementary extension of M, Z(M ′ ) denotes the interpretation in M ′ of the formula defining Z in M. M̄ denotes a saturated elementary extension of M, and we also may consider elementary extensions of M̄ in which all types over M̄ are realized. Fact 1.2. Suppose that all complete types (in any sort) over M are definable. Then every complete type p over M has a unique coheir in S(M̄ ) as well as a unique heir in S(M̄ ). This applies to the situation where M = (Qp , +, ×, −, 0, 1) due to a theorem of Delon [6]. Now suppose that G is a group definable in M. The “definable” analogue of βG is the space SG (M) of all complete types over M concentrating on G. G clearly acts on SG (M) (on the left) by homeomorphisms. If all 5 types over M are definable, then E(SG (M)) coincides with SG (M) and we already have a semigroup structure, which we denote by ∗, on SG (M). It can be explicitly described as p ∗ q = tp(gh/M) where g realizes p and h realizes the unique heir of q over (M, g). It is worth mentioning the notion of a definable action of G on a compact space X. It means an action of G on X (by homeomorphisms) with the property that for each y ∈ X the map from G to X taking y ∈ X to gy is definable. Where a map φ from G to the compact space X is said to be definable if for any two closed disjoint subsets C1 , C2 of X, the preimages φ−1 (C1 ), φ−1 (C2 ) are separated by a definable (in M) subset of G. When all types over M are definable, then (SG (M), G, idG ) is the universal definable G-ambit (in analogy with βG being the universal G-ambit). Moreover some/any minimal subflow M of SG (M) will be the universal minimal definable G-flow. And the Ellis group pM, for p any idempotent in M, will be a basic invariant of the definable group G. 00 Another basic invariant of G is the compact group G(M̄)/G(M̄ )00 M , where G(M̄ )M is the smallest bounded index type-definable over M subgroup of G(M̄ ). This can also be described as the definable Bohr compactification of G. As already pointed out in [19] there is a natural surjective homomorphism from the Ellis group to the definable Bohr compactification. Newelski suggested that in tame contexts such as when T h(M) has NIP , this is actually an isomorphism. This was proved in [5] when G is definably amenable (and proved earlier in [4] when M is o-minimal and G definably amenable). In [10] we showed that SL(2, R) as a group definable in (R, +, ×) gives a counterexample. And one of the points of the current paper is that SL(2, Qp ) provides another (counter)example. In fact it will be somewhat more striking as the (definable) Bohr compactification of SL(2, Qp ) is trivial, whereas the (definable) Ellis group will be infinite. As mentioned earlier, our counterexample is different from the ones provided by Corollary 0.3 of [16], as we have G = G000 (where G = SL(2, K) which is abstractly simple modulo its finite centre). This paper will make use of some more model-theoretic machinery, around definable amenability and f -genericity, in a NIP environment (bearing in mind that T h(Qp , +, ×, −, 0, 1) has NIP , see Section 4.2 of [2] for references). So let us now assume, in addition to G being a group definable in M, that T h(M) has NIP . G is said to be definably amenable if there is a left G-invariant Keisler measure µ on G; namely µ is a map from the Boolean algebra of subsets of G definable in M to the real unit interval [0, 1], µ(∅) = 0, µ(G) = 1, µ is finitely additive, and µ(gX) = µ(X) for all definable X and g ∈ G. Now SL(2, Qp ) (as a group definable in (Qp , +, ×, −, 0, 1)) will not be definably amenable, for the same reason that SL(2, R) is not, see [13]. But its constituents in the Iwasawa decomposition will be definably amenable. Let M̄ be a very saturated elementary extension of M. By a global type of G we mean a type p(x) ∈ SG (M̄). p is strongly f -generic if every left G-translate of p is Aut(M̄ /N)-invariant (i.e. does not fork over N) for some small N ≺ M̄ depending only on p. The existence of a strongly f -generic type is equivalent to definable amenability of G ([14]). Assuming G to be definably amenable we can take as a definition of p being f -generic that Stab(p) is G(M̄ )00 , and this is implied by p being strongly f -generic. 6 (See Section 3 of [5].) There are two extreme cases for a strongly f -generic global type p(x). The first case is when p and all of its left translates are definable over some small model N, and the second case is when p and all of its left translates are finitely satisfiable over some small model M. In the second case every strongly f -generic type is finitely satisfiable in any small model, and G is what is called an f sg group (group with finitely satisfiable generics). In this f sg case, the f -generic types, strongly f -generic types, and generic types coincide. Here a (left) generic formula is one such that finitely many (left) translates cover the group, and a (left) generic type is one all of whose formulas are (left) generic. Finally let us remark that the sister paper [10] on SL(2, R) was subsequently extended in various ways; first to a larger class of semialgebraic real Lie groups, and secondly to arbitrary real closed base fields in place of R. See [15] and [26]. The p-adic version should be able to be extended similarly. Also the analysis we give in this paper could be situated in a more general environment of definable groups G in NIP structure, where G has a nice “abstract” Iwasawa decomposition. 2 Ingredients and building blocks Our model-theoretic analysis of SL(2, R) (acting on its space of types) in [10] made heavy use of the Iwasawa decomposition SL(2, R) = K · B(R)0 with K the maximal compact subgroup SO(2, R) and B(R)0 the (real) connected component of the Borel subgroup of upper triangular matrices. Note that both K and B(R)0 are connected and also have trivial intersection. The Iwasawa decomposition for SL(2, Qp ) on the other hand has the form K · B(Qp ) where K is the maximal compact SL(2, Zp ) and B is again the Borel subgroup of upper triangular 2-by-2 matrices. B(Qp ) is itself is the semidirect product of the additive and multiplicative groups of Qp (where the action is multiplication by the square). See [3]. So we start with the model-theoretic/dynamical analysis of these building blocks. 2.1 The additive and multiplicative groups If we refer to a theory T it will be T h(Qp , +, ×, −, 0, 1). Recall that Pn (x) denotes the formula saying that x is an nth power, and that T has quantifier elimination after adding predicates for all Pn . Before getting into details we recall, with references, some basic facts which will be used freely in this section and the rest of the paper. First the topology on both the standard model Qp and the saturated model K is the valuation topology. The following can be found in (or easily deduced from) Section 1 of [18] (Facts 1 to 3) and Section 2 of [2] and make use of Hensel’s Lemma. The (nonzero) nth powers form an open subgroup of finite index in the multiplicative group, and each coset contains representatives from Z even with valuation 0. It is clear that the partial type ∩n Pn (x) defines the “connected component” (K∗ )0 of the multiplicative group K∗ of K. So every translate of (K∗ )0 can be (type)-defined over Z too. We first describe the complete 1-types over the standard model M = (Qp , +, ×, −, 0, 1). 7 Lemma 2.1. The complete 1-types over M are precisely the following: (a) The realized types tp(a/M) for each a ∈ Qp . (b) for each a ∈ Qp and coset C of (K∗ )0 in K∗ the type pa,C saying that x is infinitesimally close to a (i.e. v(x − a) > n for each n ∈ N), and (x − a) ∈ C (note this implies x 6= a). (c) for each coset C as above the type p∞,C saying that x ∈ C and v(x) < n for all n ∈ Z. Proof. (i) First we observe that every nonrealized 1-type over the standard model M is either “at infinity” namely contains the formulas v(x) < n for all n ∈ Z, or is infinitesimally close to some a ∈ Qp , namely contains the formulas v(x − a) > n for all n ∈ Z. This depends on compactness of “balls” defined by v(x) ≥ n in the standard model, and is not true over a saturated model, as we remark in 2.2(iii). Next we show that each purported complete 1-type over M described in (b) is consistent. Fix a ∈ Qp . It suffices, by compactness to show that for every n, k, and coset C of the kth powers, v(x − a) > n ∧ (x − a) ∈ C has a solution. Choose an element b in C ∩ Zp (as mentioned earlier we can find one). Let r be a natural number such that rk > n. Then bprk ∈ C and v(bprk ) > n. Let x = a + bpr k, then (x − a) has value > n and is in C. A similar argument shows consistency of any type of kind (c). Note that for any complete type p(x) over M and any a ∈ M, p has to choose some coset C of (K ∗ )0 such that “(x − a) ∈ C” is in p. So it remains to show completeness of the pa,C and p∞,C . We will do the case of p0,C from (b). (The general case of (b) is similar, by expanding polynomials around a.) To show completeness of p0,C it is enough, by quantifier elimination, to show that p0,C decides each formula of the form Pn (f (x)) where f (x) is a polynomial over Qp . Suppose f (x) = ai xi + ai+1 xi+1 + · · · + am xm where ai 6= 0. Let c realize p0,C . Then c−i f (c) = ai +ai+1 c+..+am cm−i . As v(ai+1 c+..+am cm−i ) > Z, and each (multiplicative) coset of the nth powers is open, it follows that c−i f (c) and ai are in the same coset of Pn . But the coset of Pn that c−i is in is determined by c realizing p0,C . Hence the coset of Pn in which f (c) lives is also determined, by c realizing p0,C , as required. Finally we show completeness of p∞,C from (c). Consider again a formula Pn (f (x)), (with f (x) over Qp ) and we want it to be decided by p∞,C . Again let f (x) = ai xi + ... + am xm with ai 6= 0 and am 6= 0. Let c realize p∞,C . Now we consider c−m f (c) = ai ci−m + .. + am−1 c−1 + am . Now v(ai ci−m + ... + am−1 c−1 ) > Z. So again as cosets of the nth powers are open in the multiplicative group, it follows that c−m f (c) and am are in the same coset of Pn . Hence again the coset of f (c) modulo Pn is determined by c realizing p0,C . Remark 2.2. (i) The lemma shows that the definable (with parameters) subsets of Qp are precisely given by (finite) Boolean combinations of formulas x = a, v(x − a) ≥ n and (x − a) ∈ C ′ , for a ∈ Qp , n ∈ Z and C ′ a coset of the nth powers. (ii) An identical proof to the above shows that working now over the saturated model 8 M̄ = (K, ....), if p(x) ∈ S1 (M̄ ) is a nonrealized complete 1-type “at infinity”, namely containing v(x) < Γ, then for some coset C of (K∗ )0 , p is axiomatized by v(x) < Γ together with x ∈ C. Similarly if p(x) ∈ S1 (M̄ ) is nonrealized and says that v(x−a) > Γ for some a ∈ K then for some C as before p is axiomatized by x 6= a and v(x − a) > Γ together with (x − a) ∈ C. (iii) There will be nonrealized 1-types over M̄ not accounted for in (ii), but we do not have to describe them precisely for the purposes of this paper. We start with the additive group. We consider S1 (M) as a (Qp , +)-flow. We could and should write it as SGa (M) but this is too much notation. Proposition 2.3. (i) Each type p(x) ∈ S1 (M) of kind (c) is invariant (under the action of (Qp , +)), and these account for all the minimal subflows of S1 (M). (ii) The global heirs of the types in (i) are precisely the global (strongly) f -generics of (K, +), and are all definable, and invariant under (K, +). (iii) (K, +) = (K, +)0 = (K, +)00 . Proof. (i) Let a realize p∞,C . Then clearly for b ∈ Qp , v(a + b) < Z. On the other hand, for b ∈ Qp , (a + b)/a = 1 + (b/a) and note that v(b/a) > Z. As the group of nth powers in Qp is open for all n, it follows that a + b and a are in the same coset of the nth powers for all n, and so in particular a + b ∈ C. We have shown that p∞,C is fixed under addition by elements of Qp , as required. If q(x) ∈ S1 (M) is arbitrary note that the closure of the orbit (under (Qp , +)) of q always contains a “type at infinity” namely a type of kind (c). Hence the only minimal subflows of S1 (M) are those of the form {p} for p of kind (c). (ii) and (iii). Let q be a global heir of a type p∞,C of kind (c). Then q is definable over M and Stab(q) = (K, +). This already shows that that (K, +) = (K, +)00 (= (K, +)000 ) (because any global type 1-type determines a coset of (K, +)000 ). Conversely suppose q(x) ∈ S1 (M̄ ) is an f -generic. Then by what we have just said, together with the fact that (K, +) is definably amenable, since it is abelian, q must be (K, +)-invariant. We claim first that q must be a “type at infinity”. For otherwise “v(x) ≥ γ” is in q(x) for some γ in the value group Γ of K. Then for b ∈ K with v(b) < γ, q + b 6= q, a contradiction. So q is a type at infinity as claimed. By Remark 2.2(ii), q is axiomatized by v(x) < Γ together with x ∈ C for some coset C of (K∗ )0 . But then clearly q is definable over Qp and so is the heir of p∞,C . Now for the multiplicative group. SGm (M) denotes the space of complete types over M concentrating on Gm , namely all complete 1-types except for x = 0. Gm (Qp ) is just the multiplicative group (Q∗p , ×) and SGm (M) is a Gm (Qp )-flow. Proposition 2.4. (i) SGm (M) has two minimal subflows, the collection of types of kind (b) with a = 0, namely P0 = {p0,C : C coset of (K∗ )0 } and the collection of types of kind (c), namely P∞ = {p∞,C : C coset of (K∗ )0 }. (ii) The global heirs of the types of the types mentioned in (i) are precisely the global (strongly) f -generic types of Gm , all of which are definable. Moreover the orbit of each 9 such type under K∗ is closed. (iii) (K∗ )00 = (K∗ )0 . Proof. (i) Fix any p0,C ∈ P0 . Then it is clear that the closure of its orbit under Q∗p equals P0 . Likewise for P∞ . Hence P0 and P∞ are minimal subflows. On the other hand it is clear that for any q(x) ∈ SGm (M), the closure of the orbit of q under Q∗p intersects both P0 and P∞ . Whence P0 and P∞ are the only minimal subflows of SGm (M). (ii) and (iii). Fix p0,C . Let p′0,C be its (unique) global heir. Note that p′0,C is axiomatized again by v(x) > Γ and x ∈ C. It is clear that the the stabilizer of p′0,C (with respect to the action of K∗ ) is precisely (K∗ )0 , whereby p′0,C is f -generic. On the other hand the orbit of p′0,C under K∗ is precisely P0′ the collection of global heirs of the types in P0 . Hence p′0,C is also strongly f -generic. In a similar fashion the unique global heir of each ′ p∞,C is strongly f -generic. We have shown that the types in P0′ and the analogue P∞ ∗ 0 have stabilizer (K ) , are definable over M and are all strongly f -generic. This already shows that (K∗ )00 = (K∗ )0 . Bearing in mind Remark 2.2 (ii), we have shown that global types at infinity or infinitesimally close to 0 are (strongly) f -generic. It is easy to see that these are the only (strongly) f -generics; suppose γ ∈ Γ is positive, and q(x) is a global 1-type implying that −γ < v(x) < γ. We can find g ∈ (K∗ )0 with v(g) > 2γ. But then gq implies v(x) > γ, so q is not invariant under multiplication by (K∗ )0 , so could not be f -generic. Remark 2.5. (i) So note that K∗ /(K∗ )00 is Ẑ, which is not a compact p-adic Lie group. (ii) In fact the valuation homomorphism v : K∗ → Γ induces an isomorphism between K∗ /(K∗ )00 and Γ/Γ00 . Proof. We have seen above that (K∗ )00 = (K∗ )0 which is obviously the intersection of the (K∗ )n . Notice that v takes (Q∗p )n onto nZ (as v(pnk ) = nk). So v takes (K∗ )n onto nΓ, so establishes an isomorphism between K∗ /(K∗ )n and Γ/nΓ = Z/nZ. This induces an isomorphism between the inverse limit of the K∗ /(K∗ )n and Ẑ. Finally we discuss the additive and multiplicative groups of the valuation ring O. O(M) is (Zp , +), and O∗ (M) = (Z∗p , ×), where remember that O∗ is defined by v(x) = 0. These groups (Zp , +) and (Z∗p , ×) are compact groups definable in Qp , so by Corollary 2.3 of [20], (O, +) and (O∗ , ×) are f sg groups, which have been studied intensively. We record the basic facts, leaving details to the interested reader. Proposition 2.6. (i) The universal definable minimal flow of (Zp , +) is the space SO,na (M) of nonalgebraic types concentrating on O, which are precisely the types in (b) above for a ∈ Zp . (ii) The global coheirs of the types in (i) are precisely the global (strongly) f -generic types of O which coincide with the generic types of O. (iii) The orbits in SO,na (M) are indexed by the multiplicative cosets C, namely a typical orbit is of the form {pa,C : a ∈ Zp }. Remark 2.7. Compare to the case where M = (R, +, ×) and G is the circle group (SO2 , or [0, 1) with addition mod 1). The set of nonalgebraic types is the unique minimal flow 10 and there are two orbits, infinitesimal to the left, and infinitesimal to the right (of each point of G in the standard model). Proposition 2.8. (i) The universal definable minimal flow of (Z∗p , ×) is the space SO∗ ,na (M) of nonalgebraic types concentrating on O∗ , namely the types pa,C with v(a) = 0. (ii) The global (strongly) f -generic types are precisely the coheirs of these types and they coincide with the global generic types. (iii) The Z∗p -orbits in SO∗ ,na (M) are precisely the sets {pa,aC : a ∈ Z∗p } for C a coset of (K∗ )0 . 2.2 The Borel subgroup The Borel subgroup B of SL(2, −) is the group of upper triangular 2-by-2 matrices of determinant 1.   a c ∗ So B(K) is the subgroup of SL(2, K) consisting of matrices −1 where a ∈ K 0 a   a c ∈ B(K) with the and c ∈ K. There is no harm in identifying the matrix 0 a−1 pair (a, c) ∈ K∗ × K. Likewise for B(Qp ). Note that with this notation the product (a, c)(α, β) equals (aα, aβ + cα−1). Lemma 2.9. B(K)00 = B(K)0 = {(a, c) : a ∈ (K∗ )0 , c ∈ K}. Proof. B(K) maps onto K∗ with kernel (K, +). So the result follows from Proposition 2.3 (iii) and Proposition 2.4(iii). Rather than describe all the global f -generic types of B(K) we will choose one, as follows. Let C0 denote (K∗ )0 , the connected component of the multiplicative group. Then p′0,C0 , the unique global heir of p0,C0 , is a global f -generic of K∗ , and likewise p′∞,C0 , the unique global heir of p∞,C0 is a global f -generic of (K, +). Let α realize p′0,C0 and β realize p′∞,C0 such that tp(α/M̄ , β) is finitely satisfiable in M̄ . We let p¯0 = tp((α, β)/M̄) ∈ SB (M̄ ), and let p0 = tp((α, β)/M) be its restriction to M. Note that this is new notation which will be used in Section 3 too. Lemma 2.10. p¯0 ∈ SB (M̄ ) is a global (strongly) f-generic type of B(K), every (left) B(K)-translate of which is definable over M. Proof. We note first that p¯0 is left B(K)0 -invariant: let (a, c) ∈ B(K)0 , which by Lemma 2.9 means that a ∈ (K∗ )0 . Now (a, c)(α, β) = (aα, aβ + cα−1 ). As a ∈ K∗ 0 , then tp(aα/K) = tp(α/K). On the other hand, β also realizes a global f -generic type of the multiplicative group. So tp(aβ/M̄) = tp(β/M̄). Also tp(aβ/M̄ , cα−1, aα) realizes the unique heir of tp(β/M̄) so as the latter is an f -generic of the additive group which is connected, we have that tp(aβ + cα−1 /M̄ , cα−1 , aα) is an heir of tp(β/M̄). It follows from all of this that tp((aα, aβ + cα−1 )/M̄ ) = tp((α, β)/M̄) as required. 11 As tp(α/M̄) is definable over M and tp(β/M̄ , α) is the heir of tp(β/M̄) which is definable over M, then tp(α, β/M̄) is definable over M. Using a similar argument as in the first paragraph, every left B(K)-translate of p¯0 is definable over M. Corollary 2.11. (i) The B(K)-orbit of p¯0 is closed, and hence is a minimal B(K)subflow of SB (M̄ ). (ii) Let J¯ denote the B(K)-orbit of p¯0 and J the closure of the B(M)-orbit of p0 . Then the restriction to M map gives a homeomorphism between J¯ and J , and J is a minimal subflow of SB (M). (iii) J is a subgroup of (SB (M), ∗), is isomorphic to B(M̄ )/B(M̄)0 , and is the Ellis group of the dynamical system (B(M), SB (M)). Moreover p0 is an idempotent in (SB (M), ∗). Proof. (i). This follows by using the proof of Lemma 1.15 of [23]. More specifically it is proved there that in the NIP environment, a global definable f -generic type p is almost periodic by showing that in fact the orbit of p is closed. (ii). We have seen in Lemma 2.10 that every p ∈ J¯ is the unique heir of its restriction to M. Hence the restriction to M map, π, which is a continuous map between J¯ and its image π(J¯) is a bijection, hence a homeomorphism. Now it is fairly easy to see directly that π(J¯) is a minimal B(M)-subflow of SB (M), although we can also appeal to the general result Corollary 4.7 of [25] to see this. As J is a closed B(M)-subflow of π(J¯), it follows that they are equal, and we obtain all of (ii). (iii). The natural map from SB (M̄ ) to (the profinite group) B(M̄ )/B(M̄)0 , is continuous. Moreover this map induces a bijection hence homeomorphism between J¯ and B(M̄ )/B(M̄)0 . Composing with the homeomorphism between J and J¯ gives a homeomorphism θ say between J and B(M̄ )/B(M̄)0 . It is clear that this is also an isomorphism of semigroups, whereby (J , ∗) is already a group, so must concide with the Ellis group (u ∗ J , ∗) (u an idempotent of J ). As p¯0 /B(M̄)0 is the identity of B(M̄ )/B(M̄)0 by Lemma 2.9, it follows that p0 is an (in fact the) idempotent of J . 2.3 The maximal compact subgroup As already remarked a maximal compact subgroup of SL(2, Qp ) is SL(2, Zp ). We refer to this group as K and sometimes, by abuse of language, we also let K denote the defining formula. So K(M̄ ) is SL(2, O), and SK (M), SK (M̄) denote the corresponding type spaces. (The notation O for the valuation ring in the saturated model M̄ was introduced in Section 2.1.) We have the standard part map st : SL(2, O) → SL(2, Zp ) the kernel of which is (by definition) the infinitesimals. By Corollary 2.4 of [20], this kernel coincides with SL(2, O)00 . (Note that as SL(2, Zp ) is profinite, this group of infinitesimals is an intersection of definable groups, so coincides with SL(2, O)0 .) From Corollary 2.3 of [20], K is an f sg group. In particular, we have 12 Fact 2.12. (i) Left and right generic definable subsets of K(M̄ ) coincide and are all satisfiable in M. (ii) There exist left generic types in SK (M̄), which by (i) coincide with right generic types. (iii) The unique minimal K-subflow of SK (M) is the set I of generic types over M. (iv) Likewise the unique minimal subflow I ′ of SK (M̄ ) is the set of global generic types, each such global generic type q being the unique coheir of q|M ∈ I. (v) The standard part map st induces an isomorphism (in fact homeomorphism) between K(M̄ )/K(M̄)00 and SL(2, Zp ) Recall that in the current situation where all types over M are definable, we have the semigroup operation ∗ on SK (M), and I is a left ideal under ∗. From Theorem 3.8 of [22] for example, we see that the Ellis group of the action of SL(2, Zp ) on SK (M) is canonically isomorphic to K/K 00 = SL(2, Zp ). With notation as in Fact 2.12 this Ellis group is u ∗ I for some/any idempotent in I. Different choices of u give isomorphic groups and the collection of such u ∗ I partitions I. We will elaborate slightly on these basic facts. Lemma 2.13. 1. I is a two-sided ideal of (SK (M), ∗). 2. For any q ∈ I, q ∗ SK (M) is the copy of the Ellis group which contains q. Proof. 1. Let q ∈ I and p ∈ SK (M). Let b ∈ K(M̄ ) realize p and let a realize the unique coheir of q over M̄. Then tp(ab/M) realizes q ∗ p. On the other hand, tp(a/M̄ ) is right generic, whereby tp(ab/M̄ ) is also right generic, so by Fact 2.12 (iv), q ∗ p = tp(ab/M) ∈ I. 2. Again let q ∈ I. Let E ⊆ I be the copy of the Ellis group which contains q, and let q0 be an idempotent in E. Then q ∗ SK (M) = (q0 ∗ q) ∗ (SK (M)) = q0 ∗ (q ∗ SK (M)) ⊆ q0 ∗ I (using part 1.) ⊆ q0 ∗ q ∗ I ⊆ q ∗ I ⊆ q ∗ SK (M). This shows that q ∗ SK (M) = q0 ∗ I which equals E. As usual for x, y in a given group G, xy denotes the conjugate yxy −1 of x by y and the notation extends naturally to subsets X of G in place of x ∈ G. In our context G = SL(2, K) and K(M̄ ) is SL(2, O). Lemma 2.14. (K(M̄ )0 )g = K(M̄ )0 for all g ∈ SL(2, Qp ) = G(M). Proof. We know that K(M̄ )0 is the kernel of st : K(M̄) → K(M), so equal to ∩V V (M̄ ) where V ranges over open semialgebraic neighbourhoods of the identy in K(M) = SL(2, Zp ). But SL(2, Zp ) is an open (semialgebraic) subgroup of SL(2, Qp ), so K(M̄ )0 = ∩V V (M̄ ) where V ranges over open semialgebraic neighbourhoods of the identity in SL(2, Qp ). But clearly the family of open semialgebraic neighbourhoods of the identity in SL(2, Qp ) is invariant under conjugation by elements of SL(2, Qp ). Hence the lemma follows. 13 Corollary 2.15. Let g ∈ G(M̄) and t ∈ K(M̄ )0 be such that tp(g/t, M) is finitely satisfiable in M. Then tg ∈ K(M̄ )0 . If in addition, tp(t/M) is a generic type of K then so is tp(tg /M). Proof. The first sentence is fairly immediate from Lemma 2.14: if by way of contradiction tg ∈ / V (M̄ ) for some open semialgebraic neighbourhood of the identity of K(M), then there is g1 ∈ G(M) such that tg1 ∈ / V (M̄), contradicting Lemma 2.14. The second sentence follows from the fact that the set of generic types in SK (M) is closed. 3 SL2(Qp) We use the above material to describe the minimal definable universal subflow of SL(2, Qp ) as well as its Ellis group. We first identify the minimal subflow, see Theorem 3.4 below. 3.1 Minimal subflow of (G(M), SG(M)) The Iwasawa decomposition of SL(2, Qp ) is B(Qp ) · SL(2, Zp ), namely every element of SL(2, Qp ) can be written as a product ht with h ∈ B(Qp ) and t ∈ SL(2, Zp ) (and also as a product t1 h1 with t1 ∈ SL(2, Zp ) and h1 ∈ B(Qp )). However, in contradistinction to the Iwasawa decomposition for real Lie groups, there is a large intersection of the constituents; namely B(Qp ) ∩ SL(2, Zp ) = B(Zp ) =    a c ∗ |a ∈ Zp and c ∈ Zp . 0 a−1 We recall the notation from the previous sections: K(M) = SL(2, Zp ) is the maximal compact subgroup of SL(2, Qp ), and I is the unique minimal subflow of the flow (K(M), SK (M)). We fix a generic type q0 ∈ SK (M) which concentrates on K 0 . p0 is the restriction to M of the global f -generic type p¯0 of B(K), and J is the minimal subflow of (B(M), SB (M)) containing p0 , as in subsection 2.2. Lemma 3.1. I ∗ J ⊆ SG (M) ∗ q0 ∗ p0 . Proof. We have to show that for any q1 ∈ I and p1 ∈ J , there is s ∈ SG (M) such that s ∗ q0 ∗ p0 = q1 ∗ p1 . Let p′ ∈ SB (M) be such that p′ ∗ p0 = p1 . (Because p0 , p1 ∈ J which is a minimal subflow of SB (M) so of the form SB (M) ∗ p0 .) Now let s = q1 ∗ p′ . Then s ∗ q0 ∗ p0 = q1 ∗ p′ ∗ q0 ∗ p0 = tp(t0 hth0 /M) = tp(t0 th hh0 /M) 14 where t0 realizes q1 , h realizes the (unique) heir of p′ over M, t0 , t realizes the unique heir of q0 over M, t0 , h and h0 realizes the unique heir of p0 over M, t0 , h, t. We may assume that t0 , h, t are in SL(2, K) and that h0 realises the unique heir of p0 over M̄ . By Lemma 2.10, tp(hh0 /M̄) is definable over M, and note that tp(hh0 /M) = p′ ∗p0 = p1 . On the other hand, by Corollary 2.15, th ∈ K 0 . As q1 ∈ I and t0 realizes the unique coheir of q1 over M, th , we have that tp(t0 th /M) = q1 . Hence tp(t0 th hh0 /M) = q1 ∗ p1 as required. Lemma 3.2. SG (M) ∗ q0 ∗ p0 = cl(I ∗ J ) Proof. The previous lemma together with the fact that SG (M) ∗ q0 ∗ p0 is closed shows that the RHS is contained in the LHS. For the converse, we will show that the G(M) orbit of q0 ∗ p0 is contained in I ∗ J which suffices, by taking closures, to see that the LHS is contained in the RHS. So let g ∈ G(M) and write g = th with t ∈ K(M) = SL2 (Zp ) and h ∈ B(M) = B(Qp ). Then (th)(q0 ∗ p0 ) = tq0h ∗ hp0 = tp(tth0 hh0 /M) where t0 realizes q0 and h0 realizes the unique heir of p0 over M, t0 . By the choice of J , tp(hh0 /M) ∈ J . Clearly (or by 2.15), tp(th0 /M) ∈ I, as is tp(tth0 /M). Now as tp(h0 /M, t0 ) is an heir of its restriction to M, also tp(hh0 /M, tth0 ) is an heir of its restriction to M, so tp(tth0 hh0 /M) ∈ I ∗ J , so by the displayed equation above g(q0 ∗ p0 ) ∈ I ∗ J , as required. Lemma 3.3. SG (M) ∗ q0 ∗ p0 ⊆ SK (M) ∗ J . Namely every s ∗ q0 ∗ p0 (with s ∈ SG (M)) is of the form r ∗ p with r ∈ SK (M) and p ∈ J . Proof. Let s = tp(th/M) where t ∈ K(M̄ ) = SL(2, O) and h ∈ B(M̄ ) = B(K). Then s ∗ q0 ∗ p0 = tp(tht0 h0 /M) = tp(tth0 hh0 /M) where t0 realizes the unique heir of q0 over (M, t, h) and h0 realizes the unique heir of p0 over M̄ , namely p¯0 . Again, tp(hh0 /M̄ ) is definable over M (by Lemma 2.10), and tp(hh0 /M) ∈ J (by Corollary 2.11). Moreover by 2.15, th0 ∈ K and so also tth0 ∈ K. Thus tp(tth0 hh0 /M) ∈ SK (M) ∗ J as required. Theorem 3.4. (i) cl(I ∗ J ) is a minimal subflow of the flow (G(M), SG (M)). (ii) Moreover q0 ∗ p0 is an idempotent in this minimal flow. Proof. (i). By Lemma 3.2, cl(I ∗ J ) is a G(M)-flow. As any point in cl(I ∗ J ) is of the form s ∗ q0 ∗ p0 by Lemma 3.2, and the closure of the G(M)-orbit of this s ∗ q0 ∗ p0 is precisely SG (M) ∗ s ∗ q0 ∗ p0 , it suffices to prove: 15 Claim. For any s ∈ SG (M), I ∗ J ⊆ SG (M) ∗ s ∗ q0 ∗ p0 . Proof of claim. Fix s ∈ SG (M). By the previous lemma, let q ′ ∈ SK (M), and p1 ∈ J be such that (1) s ∗ q0 ∗ p0 = q ′ ∗ p1 , and note that by Lemma 2.10 and Corollary 2.11 (ii) the unique global heir of p1 is a strong f -generic of B every translate of which is definable over M. We can easily find q1 ∈ I such that (2) q1 ∗ q ′ ∈ K 0 (in the obvious sense that some/any realization is in K 0 ). Now, let q ∈ I and p ∈ J and we want to show that q ∗ p ∈ SG (M) ∗ s ∗ q0 ∗ p0 . Let p′ ∈ SB (M) be such that: (3) p′ ∗ p1 = p, where p1 is as in (1). Now we compute q ∗ p′ ∗ q1 ∗ q ′ ∗ p1 . Let a realize q, b realize the unique heir of p′ over (M, a), c realize the unique heir of q1 ∗ q ′ over (M, a, b) and d realize the unique heir of p1 over M̄ (so in particular over (M, a, b, c)). Then (4) q ∗ p′ ∗ q1 ∗ q ′ ∗ p1 = tp(abcd/M) = tp(acb bd/M) Now by the property of p1 in (1), tp(bd/M̄ ) is definable over M. In particular (using (3)) bd realizes the unique heir of p over (M, acb ). On the other hand, by 2.15 and (2), cb ∈ K 0 (M̄ ). As tp(c/M, a, b) is definable over M, and tp(b/M, a) is definable over M, tp(a/M, b, c) is finitely satisfiable in M (and moreover realizes the unique coheir over (M, b, c) of q, as all types over M have unique heirs). As the stabilizer (inside K) of the global coheir of q is K 0 , it follows that tp(acb /M) = q. So we conclude that (5) tp(acb bd/M) = tp(acb /M) ∗ tp(bd/M) = q ∗ p. By (4) and (5) q ∗ p = r1 ∗ (q ′ ∗ p1 ) where r1 = q ∗ p′ ∗ q1 ∈ SG (M). So by (1) q ∗ p = r1 ∗ s ∗ q0 ∗ p0 giving the claim. End of Proof of claim. This finishes the proof of (i). (ii) is an easy computation, bearing in mind the techniques above, which we carry out below. We want to show that q0 ∗ p0 ∗ q0 ∗ p0 = q0 ∗ p0 The left hand side is tp(tht0 h0 /M), where t and t0 realize q0 , h and h0 realize p0 , and tp(t/M, h0 , t0 , h0 ) is the coheir of q0 etc. We will slightly adapt the proof of Lemma 3.3. First rewrite this left hand side as tp(t(th0 )hh0 /M). Conclude from 2.15 that th0 ∈ K(M̄ )0 . But K(M̄ )0 is the stabilizer of the unique global coheir q¯0 of q0 , whereby t(th0 ) realizes q0 . On the other hand, we may assume that h0 realizes the global heir p¯0 of p0 (and that t, h, t0 are in M̄). As the stabilizer of p̄0 is B(M̄ )0 which contains h it follows that hh0 also realizes p¯0 . Putting it together we see that tp(t(th0 )hh0 /M) = q0 ∗ p0 , as required. Note that from Theorem 3.4 and the discussion in subsection 1.2, we have identified the universal definable minimal flow of SL(2, Qp ). Moreover we have shown that q0 ∗ p0 is almost periodic and idempotent. 16 3.2 The Ellis group Let M denote the minimal G(M)-flow SG (M) ∗ q0 ∗ p0 = cl(I ∗ J ). The Ellis group attached to the flow (G(M), SG (M)) is then the group (q0 ∗ p0 ∗ M, ∗) which we aim to describe explicitly. Remember that the intersection of K(M) (i.e. SL(2, Zp )) and B(Qp ) is B(Zp ). Lemma 3.5. Let h realize p0 . Let t ∈ SL(2, Zp ). Then • if t ∈ B(Zp ), then p0 t = t tp(h′ /M), for some h′ ∈ B(K)0 ∩ dcl(h, M) • if t ∈ / B(Zp ), then p0 t = tp(t′ h′ /M), where t′ ∈ SL(2, O)0 ∩ dcl(h, M) and h′ ∈ B(K) ∩ dcl(h, M). Proof. The first case is immediate as p0 (x) implies x ∈ B(K)0 , B(K)0 is normal in B(K) and t ∈ B(K).   u1 u2 For the second case: Let t = such that u3 6= 0. Let h = (a, c) realize p0 . u3 u4 Then      1 0 au1 + cu3 au2 + cu4 au1 + cu3 au2 + cu4 ′ ′ = a−1 u3 ht = −1 = t h . a−1 u3 a−1 u4 1 0 (au + cu ) 1 3 au1 +cu3 −1 u3 ) = v(a−1 u3 ) − Since v(c) < dcl(v(a), Z), we have that au1 + cu3 6= 0 and v( aua1 +cu 3 −1 u 3 v(au1 + cu3 ) = v(a−1 u3 ) − v(cu3 ) > Z. So st( aua1 +cu ) = 0. This implies that t′ = 3   1 0 ∈ SL(2, O)0 . Clearly t′ and h′ are definable over M, h. a−1 u3 1 au1 +cu3 Lemma 3.6. q0 ∗ p0 ∗ M = q0 ∗ J . Proof. We first prove that q0 ∗ J ⊆ q0 ∗ p0 ∗ M. Let q0 ∗ p be in the left hand side, namely p ∈ J . Using Corollary 2.11(iii), we have that q0 ∗ p = q0 ∗ p0 ∗ p, and is clearly in I ∗ J , so in M. So it suffices to show that q0 ∗ p0 ∗ p = q0 ∗ p0 ∗ q0 ∗ p0 ∗ p which is immediate as q0 ∗ p0 is an idempotent (Theorem 3.4). We now want to show that q0 ∗ p0 ∗ M ⊆ q0 J By Lemma 3.3 it suffices to prove that q0 ∗ p0 ∗ SK (M) ∗ J ⊆ q0 ∗ J Let q ∈ SK (M) and p ∈ J . Let r ∈ SL(2, Zp ) be the standard part of q and let q ′ = r −1 q. So q ′ ∈ K 0 , and q0 ∗ p0 ∗ q ∗ p = q0 ∗ p0 ∗ r ∗ q ′ ∗ p 17 Now we have two cases: Case (i). r ∈ B(Zp ). Let t realize q0 , h realize the heir of p0 over (M, t), t′ realize the heir of q ′ over (M, t, h) , with t, h, t′ ∈ G(M̄ ) and let h′ realizes the global heir of p. By the first part of Lemma 3.5 and our case analysis, hr = rh1 with h1 ∈ B(K)0 ∩ dcl(M, h). So q0 ∗ p0 ∗ r ∗ q ′ ∗ p h = tp(trh1 t′ h′ /M) = tp(trt′ 1 h1 h′ /M) h = tp(trt′ 1 /M) ∗ tp(h1 h′ /M). But t′h1 is in K 0 (as h1 ∈ dcl(M, h) and we can use Corollary 2.15), and tr realizes the unique coheir over (M, t′h1 ) of the generic type q0 r of K, whereby tp(trt′h1 /M) = q0 r. As before tp(h1 h′ /M) = p. We have shown so far that q0 ∗ p0 ∗ q ∗ p = q0 r ∗ p = q0 ∗ rp. As r is assumed to be in B(Zp ) we see that rp ∈ J too. So q0 ∗ rp ∈ q0 ∗ J as required. Case (ii). r ∈ SL(2, Zp ) \ B(Zp ). By the second part of Lemma 3.5, p0 r = tp(t0 h0 /M) with t0 ∈ SL(2, O)0 , h0 ∈ B(K) and both t0 , h0 ∈ dcl(M, h), for h = (a, c) realizing p0 . Now choose t realizing the unique coheir of q0 over (M, h) and t′ realizing the unique heir of q ′ over (M, t, h), with t, t′ , h in G(M̄). Now let h′ realize the unique heir of p over M̄ . So, by the remarks above, ′ q0 ∗ p0 ∗ r ∗ q ′ ∗ p = tp(tt0 h0 t′ h′ /M) = tp(tt0 (h0 t′ h−1 0 )h0 h /M). Now, as t0 and t′ are in SL(2, O)0 and using Corollary 2.15, we see that t0 (h0 t′ h−1 0 ) ∈ SL(2, O)0 , and as t realizes the unique coheir of q0 over these elements, tp(tt0 h0 t′ h−1 0 /M) = ′ q0 . On the other hand, now standard arguments give that tp(h0 h /M̄) is the unique ′ global heir of tp(h0 h′ /M) ∈ J . Hence tp(tt0 (h0 t′ h−1 0 )h0 h/M) is of the form q0 ∗ p for some p′ ∈ J , and Case (ii) is complete. We have shown that the Ellis group attached to the flow (G(M), SG (M)) is q0 ∗ J . Theorem 3.7. The map from J to q0 ∗J which takes p to q0 ∗p, is a group isomorphism between (J , ∗) and (q0 ∗ J , ∗). Proof. We first show that for p, p′ ∈ J , q0 ∗ p = q0 ∗ p′ iff p = p′ . Suppose that q0 ∗ p = q0 ∗ p′ . Hence there are realizations t, t′ of q0 , h of p and h′ of p′ such that th = t′ h′ . Note that t and t′ are both in SL(2, O)0 . So (t′ )−1 t = h′ h−1 ∈ SL(2, O)0 ∩ B(K). But SL(2, O)0 ∩ B(K) is easily seen to be B(O)0 which is contained in B(K)0 . This shows that h and h′ are in the same coset of B(K)0 in B(K), which implies that p = p′ . So we have shown that the map taking p ∈ J to q0 ∗p establishes a bijection between J and q0 ∗ J . So the Theorem will be established after proving that for p, p′ ∈ J , q0 ∗ p ∗ q0 ∗ p′ = q0 ∗ p ∗ p′ 18 Claim. Let p, p′ ∈ J . Then p ∗ q0 ∗ p′ = tp(t0 /M) ∗ p ∗ p′ for some t0 ∈ SL(2, O)0 . Proof of claim. Let h0 realize p in M̄ , let t1 ∈ SL(2, O)0 realize the unique heir of q0 over M, h0 , and let a realize the unique global heir of p. Then p ∗ q0 ∗ p = tp(h0 t1 a/M) = tp((h0 t1 h−1 0 )h0 a/M). 0 0 Put t0 = h0 t1 h−1 0 which is in SL(2, O) (i.e. in K(M̄ ) with earlier notation). By Lemma 2.10 and Corollary 2.11, tp(h0 a/M̄ ) is definable over M, and clearly h0 a realizes p ∗ p′ . So t0 h0 a realizes tp(t0 /M) ∗ (p ∗ p′ ) = tp(t0 /M) ∗ p ∗ p′ , proving the claim. Now fix p, p′ ∈ J . Let t0 be given by the claim, and let h0 , h1 be realizations of p, p′ respectively in M̄ such that h0 realizes the unique heir of p over M, t0 and h1 realizes the unique heir of p′ over M, t0 , h0 . So t0 h0 h1 realizes tp(t0 /M) ∗ p ∗ p′ = p ∗ q0 ∗ p′ (by the claim). Let t realize the unique global coheir q0′ of q0 . As Stab(q0′ ) = SL(2, O)0 , it follows that q0′ t0 = q0′ . Hence, putting everything together, q0 ∗ p ∗ q0 ∗ p′ = q0 ∗ tp(t0 /M) ∗ p ∗ p′ = tp(tt0 /M) ∗ p ∗ p′ = q0 ∗ p ∗ p′ as required. This completes the proof of Theorem 3.7. Corollary 3.8. The Ellis group attached to the action of SL(2, Qp ) on its type space is Ẑ (as an abstract group). Proof. By 2.11, (J , ∗) is isomorphic to B(K)/B(K)0 which is in turn isomorphic to Ẑ by Remark 2.5 and Lemma 2.9. Question 3.9. What is the (definable) generalized Bohr compactification of SL(2, Qp )? Does it already coincide with the (definable) Ellis group identified in Theorem 3.7. Explanation. The generalized Bohr compactification was defined in [12] as a certain quotient of the Ellis group, namely by the intersections of the closures of the neighbourhoods of the identity in the so-called τ -topology on the Ellis group. This account of the generalized Bohr compactification was discussed in [16] and studied further there in the model-theoretic context. 4 The action of SL(2, Qp) on the type space of the projective line over Qp Let P1 (Qp ) denote the projective line over Qp , naturally a definable set in M. SP1 (M) denotes the space of complete types over M which concentrate on the definable set P1 (Qp ). The usual action of SL(2, Qp ) on P1 (Qp ) extends to an action on SP1 (M). We will study this action and observe that the collection of nonalgebraic types in SP1 (M) is a minimal proximal SL(2, Qp ) flow. We begin with some prequisites concerning projective space and compatibilities with our earlier notation. 19   a P (Qp ) is defined to be the set of equivalence classes of vectors 0 of elements   a1   a λa0 for all of Qp , not both zero, under the equivalence relation given by 0 ∼ a1 λa1 λ ∈ Qp , λ 6= 0. P1 (Qp ) is of course interpretable in the structure M, and we can definably identify it   a with a ∈ Qp and denoting the ∼-class with Qp ∪ {∞} by identifying the ∼-class of 1   1 by ∞, the point at infinity. Here ∞ is some fixed tuple from M. From now on of 0   a instead of its ∼-class. we may write b Note that P1 (Qp ) is a p-adic analytic  manifold  via the natural bijections φ1 : Qp −→ 0 1 P(Qp ) \ and φ2 : Qp −→ P(Qp ) \ which give P(Qp ) a manifold structure. 1 0 This p-adic manifold structure is also definable in the structure M. But we will be mainly interested in P1 (Qp ) as a definable set in M. P1 (K) denotes the obvious thing, and in fact we can consider P1as a formula of M.    inthe language ax + by x a b ; this action = · The standard action of G(Qp ) on P1 (Qp ) is: cx + dy y c d   a b is invertible. Moreover the same formula gives an action of is well-defined since c d G(K) on P1 (K). In any case we obtain an action of G(Qp ) on the compact space SP1 (M) which is a definable action as discussed earlier.   1 Remark 4.1. • The stabilizer of is B(Qp ). 0 1 • The quotient space G(Qp )/B(Qp ) is homeomorphic to P(Qp ) via:    a/c   if c 6= 0      1 a b . /B(Qp ) 7→   c d   1   if c = 0  0 Remark 4.2. We have given above a definable identification of P1 (Qp ) with Qp ∪ {∞}. The same thing identifies P1 (K) with K ∪ {∞}. Hence the type space SP1 (M) identifies with the space S1 (M) of complete 1-types over M, together with the point ∞, which is considered as a realized type. Note that with notation from Section 2.1, the 1-types over M of the form p∞,C will be the types of elements of P1 (K) which are infinitesmially close to the point ∞ (with respect to the p-adic manifold topology discussed earlier). Definition 4.3. For p ∈ SG (M) and q in SP1 (M) we define p ∗ q as tp(g · b/M) where b realizes q and g realizes the unique coheir of p over (M, b). 20 Remark 4.4. (i) If p1 , p2 ∈ SG (M) and q ∈ SP1 (M) then (p1 ∗ p2 ) ∗ q = p1 ∗ (p2 ∗ q). (ii) Let π be the map from SL(2, Qp ) onto P1 (Qp ) defined implicitly in 4.1, extended naturally to a map between the respective type spaces. Then for p1 , p2 ∈ SG (M), p1 ∗ π(p2 ) = π(p1 ∗ p2 ). (iii) For any q ∈ SP1 (M) the closure of the G(M)-orbit G(M) · q is precisely {p ∗ q : p ∈ SG (M)}. We now use some notation from earlier sections. Specifically p0 ∈ SB (M) ⊂ SG (M) and q0 ∈SK (M) (M) are specific f -generic types.  Let  ⊂ SG a /Qp ∈ SP1 (Qp )| v(a) < n : n ∈ Z . T P∞ = tp 1 So T P∞ is the infinitesimal neighbourhood of ∞ in SP1 (M) (with the topology coming from the manifold topology on P1 (Qp )), minus the point ∞ itself. Then Lemma 4.5. For every q ∈ SP1 (Qp ), • if q 6= ∞, then p0 ∗ q ∈ T P∞ . • p0 ∗ ∞ = ∞. SP1 (Qp )   a and q 6= ∞. Let be a realization of q and 1 • Suppose that q ∈   b c a realization of p0 such that tp(a/Qp , b, c) is the heir of tp(a/Qp ). h= 0 b−1 Then      a b c /Qp = tp(ab2 + bc/Qp ). · p0 ∗ q = tp 1 0 b−1 Proof. If v(a) > n for some n ∈ Z, since v(c) < dcl(Z, b) we have v(bc) < v(ab2 ) and thus v(ab2 + bc) = v(bc) < Z; if v(a) < Z, then v(ab2 ) < v(bc) since tp(v(a)/Z, v(b), v(c)) is an heir of tp((v(a)/Z), so v(ab2 + bc) = v(ab2 ) < Z. This implies that p0 ∗ q ∈ T P∞ . • As B(K) stabilizes ∞ and p0 ∈ SB (Qp ), we have that p0 ∗ q = q where q = ∞. Lemma 4.6. For every q ∈ T P∞ , we have q0 ∗ q = q0 ∗ ∞.   c be a realization of Proof. Suppose that q ∈ T P∞ and q 6= ∞. Let 1          1 1 0 1 0 1 1 0 . So −1 = . Since v(c) < Z, st · −1 −1 c 0 1 1 0  c 1  c 1 0 1 p = tp( −1 /Qp ). Then q = p ∗ ( ). So c 1 0     1 1 q0 ∗ q = q0 ∗ (p ∗ ( )) = (q0 ∗ p) ∗ ( ). 0 0 21   c = q. Then 1  0 ∈ K 0 . Let 1 Since q0 is generic in SK (M) and p is realized by some element from K 0 , we have q0 ∗ p = q0 . By Lemma 4.5 and Lemma 4.6, we have    1 . Theorem 4.7. (q0 ∗ p0 ) ∗ (SP1 (Qp )) = q0 ∗ 0 Corollary 4.8. The set of nonalgebraic types in SP1 (M) is a minimal proximal SL(2, Qp )flow. Proof. Firstly the set SP1 ,na (M) of nonalgebraic types in SP1 (M) is closed and SL(2, Qp )invariant. Secondly (see Remark 4.4(iii)), any minimal SL(2, Qp )-subflow of SP1 ,na (M),   1 . We have shown so far is closed under SG (M)∗, so by Theorem 4.7 contains q0 ∗ 0 that SP1 (M) has a unique minimal subflow which is the closure of the orbit of q0 ∗∞ and that this minimal subflow is proximal. It remains to see that SP1 ,na (M) is a minimal subflow. It is clearly closed and SL(2, Qp )-invariant. Note that the unique minimal subflow of SP1 (M) contains a minimal subflow with respect to the multiplicative group, which by Proposition 2.4 consists either of the complete nonalgebraic 1-types over M which are “infinitesimally close” to 0, or the complete nonalgebraic 1-types over M which are “at infinity”, namely what we called above the infinitesimal neighbourhood of ∞ in SP1 ,na (M). As SL(2, Qp ) acts transitively on P1 (Qp ), it follows that for every a ∈ P1 (Qp ), the set of complete nonalgebraic 1-types over M infinitesimally close to a is included in the unique minimal subflow of SP1 ,na (M). But this accounts for all of SP1 ,na (M) which is therefore minimal is claimed. Question 4.9. (i) Is SP1 ,na (M) the universal minimal proximal definable SL(2, Qp ) flow? (ii) Is SP1 ,na (M) a strongly proximal SL(2, Qp )-flow? Explanation. (i) The universal minimal proximal definable flow exists and will be a minimal proximal SL(2, Qp )-flow which is a “homomorphic image” of the universal minimal definable flow, and universal such. (ii) Strong proximality of (X, G) means that the action of G on the space of Borel probability measures on X is proximal. In our context, (SP1 ,na (M), SL(2, Qp )), the action will be definable. See Proposition 6.3 of [16]. We guess that the answer to (ii) is positive. References [1] J.Auslander, Minimal flows and their extensions, North Holland, Amsterdam, 1988. [2] Luc Belair, Panorama of p-adic model theory, Ann. Sci. Math. Quebec, 36(1) January 2012 22 [3] F. Bruhat and J. Tits, Groupes réductive sur un corps local, Publ. Math. 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