On compactifications and the topological dynamics of definable groups

We discuss definable compactifications and topological dynamics. For G a group definable in some structure M, we define notions of"definable"compactification of G and"definable"action of G on a compact space X (definable G-flow), where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as G*/G*00_M and the universal definable G-ambit as the type space S_{G}(M). We also prove existence and uniqueness of"universal minimal definable G-flows", and discuss issues of amenability and extreme amenability in this definable category, with a characterization of the latter. For the sake of completeness we also describe the universal (Bohr) compactification and universal G-ambit in model-theoretic terms, when G is a topological group (although it is essentially well-known).


Introduction and preliminaries
Given a topological (Hausdorff) group G, a compactification of G is a pair (H, f ) where H is a compact topological group and f : G → H a continuous homomorphism with dense image. There is a universal such compactification, called the Bohr compactification. Let us note immediately that a compactification of the topological group G is a special case of continuous action of G on a compact space X, where X has a distinguished point x 0 with dense orbit under G (a so-called G-ambit.) Again there is a universal G-ambit. There is reasonably compehensive account of abstract A big influence on this paper is work of Newelski (for example [8], [9]) on trying to use the machinery of topological dynamics to extend stable group theory to tame unstable contexts. The present paper is quite soft, aiming partly at putting Newelski's ideas into a formal framework, definable, or tame, topological dynamics. On the other hand the treatment of f sg groups in [13] and the case analysis of SL(2, R) in [4] from this point of view, are rather harder.
In the remainder of this introduction, we recall some very basic model-theoretic notions and constructions. A good reference for the kind of model theory in the current paper is [14] as well as [6]. Whenever we talk about a topological space we assume Hausdorffness.
We fix a complete theory T in language L, a model M of T and a very saturated elementary extension M * of M, for example κ-saturated of cardinality κ where κ > 2 |M |+|L| . For X a definable set in M * , definable over M (or even a set definable in M), S X (M) denotes the Stone space of complete types over M which concentrate on (the formula defining) X. By a type-definable over M set in M * we mean the common solution set in M * of a collection of formulas over M, equivalently an in-tersection of sets, definable in M * over M. Sometimes L M denotes the language L expanded by constants for elements of M.
We first recall the logic topology on bounded hyperdefinable sets (in a saturated model).
Definition 1.1. Let X be a definable set in the structure M * , definable with parameters from M.
(i) Suppose E is a type-definable over M equivalence relation with a bounded number of classes, that is < κ (equivalently ≤ 2 |M |+|L| ) many, and π : X → X/E the canonical surjection. The logic topology on X/E is defined as follows: (ii) Suppose C is a compact space, and f : X → C a map. We say that f is definable over M, if for any closed subset D of C, f −1 (D) is type-definable over M.
Lemma 1.2. Let X, E be as in Definition 1.1 and C be a compact space.
(i) The set X/E equipped with the logic topology is a compact space, and π : X → X/E is definable over M.
(ii) Conversely, suppose f : X → C is an M-definable map. Then: , for x, y ∈ X, is typedefinable over M and bounded, (c) f induces a homeomorphism between X/E equipped with the logic topology and D.
(iii) Let again f : X → C be M-definable. Then f factors through the tautological map a → tp(a/M) from X to S X (M), and C is equipped with the quotient topology.
(i) Let N be a normal, type-definable over M, bounded index subgroup of G * . Then G * /N equipped with the logic topology is a compact group.
(ii) There is a smallest, type-definable over M, subgroup of G. It is called (G * ) 00 M , and is normal in G * .
We also recall: (ii) If A ⊇ M, and q(x) ∈ S(A), then q is said to be finitely satisfiable in M (or equivalently a coheir of q|M) if every formula in q is satisfied by some element or tuple from M.
(iii) If A ⊇ M and q(x) ∈ S(A), then q is said to be an heir of q|M if for any L M -formula ϕ(x, y) such that ϕ(x, a) ∈ q for some a ∈ A, there is m ∈ M such that ϕ(x, m) ∈ q.  The property, for a given theory T , that all complete types over all models are definable (equivalently all complete types over a sufficiently saturated model are definable), is very strong, and equivalent to stability of T . On the other hand there are certain important structures M over which all complete types are definable, even though Th(M) is unstable. Examples are (R, +, ·) and (Q p , +, ·). There is also a class of first order theories, properly containing the class of stable theories, for which certain "mild" expansions of arbitrary models have the property that all types over them are definable: T is (or has) NIP , if (working in a saturated model M * ) for any L-formula ϕ(x, y), indiscernible sequence (a i : i < ω) and b ∈ M * , the truth value of ϕ(a i , b) stabilizes as i → ∞. Given a model M of any theory T , by M ext we mean the structure whose universe is M and which has predicates for all externally definable subsets of M n : where X ⊆ M n is said to be externally definable, if there is a formula ψ(x) possibly with parameters from M * such that X = {a ∈ M n : M * |= ψ(a)}. Note that M ext has (essentially) constants for all elements. The following is due to Shelah [17], but see also [12] for a simplified account. In fact Shelah suggests, in Thesis 0.5 of [18], that in studying an NIP theory T , one should from the start work with the class of externally definable sets. This in some sense provides some justification for our "definability of types" assumption when we introduce "definable topological dynamics" in Section 3.

The topological case
We give a brief model-theoretic description of the topological case. We refer the reader to [19] for a nice treatment of topological dynamics.
Let G be a topological group. As mentioned in the introduction G has a unique "universal compactification", called Bohr compactification. Here is a summary of a model-theoretic account. Let M be a structure whose universe is G, and has a symbol for the group operation, and, for each open subset of G a predicate symbol P U whose interpretation in M is U (and maybe more). We suppose the language L of M has cardinality at most 2 |M | . Let M * be a very saturated elementary extension of M and G * the corresponding group. Let U * denote the interpretation of P U in M * . The following proposition immediately follows from [3, Theorem 5.6].
Proposition 2.1. Let N be the smallest subgroup of G * which has bounded index, and is an intersection of some of the U * 's. Then: is continuous and is the universal compactification of the topological group G.
Now we consider topological dynamics. G remains a topological group. By a G-flow (X, G) we mean a continuous action of G on a compact space X (i.e. an action of G on X such that the corresponding function G × X → X is continuous). Note that for X a compact space, the space Homeo(X) of homeomorphisms of X equipped with the compact-open topology, is a compact group. Moreover a G-flow (X, G) is precisely given by a continuous homomorphism from G to Homeo(X). Note that if x ∈ X, then the map taking g ∈ G to g·x ∈ X is continuous. By a G-ambit (X, There is a universal G-ambit, sometimes called S(G), with distinguished point e say: for every G-flow (X, G) and p ∈ X there is a unique map of G-flows from S(G) to X which takes e to p. The universal G-ambit is unique, by definition. The following model-theoretic (nonstandard analytic) account of the universal G-ambit is easily seen to follow from the account of S(G) as the "Samuel compactification" of the uniform space (G, R) where R is the canonical right uniformity on G (see [19,Section 2]). The basic entourages of R are {(x, y) ∈ G 2 : xy −1 ∈ V }, where V ⊆ G range over open sets containing the identity 1 of G.
Proposition 2.2. Let G be a topological group, and M a structure as described at the beginning of this section. Let E be the finest bounded, type-definable over M equivalence relation on G * which satisfies the following condition: whenever g, h ∈ G * and gh −1 ∈ U * for every U which is a neighbourhood of the identity, then E(g, h).
Then E is G-invariant, and G * /E with the logic topology and distinguished point 1/E is the universal G-ambit.
Proof (sketch). S(G) is the completion of G with respect to the finest precompact uniformity U which is coarser than R. Such a uniformity corresponds to a typedefinable equivalence relation E on G * . Since U is precompact, E has boundedly many classes. Also U is G-invariant, as R is. The completion of G with respect to U is exactly G * /E with the logic topology.
We give a few additional remarks. The universal G-ambit S(G) has a canonical semigroup structure (which in the proof of Theorem 2.1 in [19] is seen to follow directly from the universality property of S(G)). This semigroup structure is in turn used to prove the uniqueness of the universal minimal G-flow: So uniqueness of the universal minimal G-flow is not built into the definition, but is nevertheless true. Computation of the universal minimal flow M(G) for various topological groups is an important enterprise. We have been told that for no locally compact noncompact topological group G, has M(G) been explicitly described. The topological group G is said to be (extremely) amenable if for every G-flow (X, G), there is a G-invariant Borel probability measure on X (X has a G-invariant point). Clearly it suffices for the universal G-ambit to have this property.

The definable case
The aim in this main section of the paper is to try to give appropriate analogues of compactification and action on a compact space, for definable groups rather than topological groups. Many of our formulations can possibly be improved or modified. We also ask a few questions. We fix a first order structure M, not necessarily saturated. By a definable set in M we mean a subset Y of M n definable with parameters from M. M * denotes a very saturated elementary extension of M. For Y a definable set in M, we often use Y * to denote the interpretation in M * of the formula which defines Y in M.
We first give an analogue of Definition 1.1(ii) in the nonsaturated case: Definition 3.1. Let Y be a definable set in M and C a compact space. By a definable map f from Y to C we mean a map f such that for any disjoint closed subsets Claim. ϕ∈p f (ϕ(M)) is a singleton in C. We start with the rather easy case of group compactifications. Proof. Let f : G → C be a definable compactification of G. By Lemma 3.2(i), f lifts uniquely to an M-definable map f * from G * onto C. We claim that f * is a group homomorphism. The proof of Lemma 3.2(i) shows that for any a ∈ G * , f * (a) = = f * (a) · f * (b). This is enough to show that f * is a homomorphism. So the kernel of f * is a normal type-definable (over M) subgroup of G * of bounded index. Also by 1.2, the topology on C coincides with the logic topology (on G * /N). So if (G * ) 00 M is the smallest type-definable over M subgroup of G * of bounded index, then we have a canonical surjection map π : G * /(G * ) 00 M → C = G * /N, which is clearly a continuous homomorphism and commutes with the canonical homomorphisms from G. Now for definable group actions. Let again G be a group definable in a structure M. So we have the action of G on the type space S G (M). G, as a subgroup of G * , also acts on the left on G * . Suppose E is a bounded, type-definable over M, G-invariant equivalence relation on G * . Then G acts on G * /E (a compact space equipped with the logic topology), and this action factors through the action of G on S G (M). Moreover if 1 is the identity element of G then the orbit of 1/E in G * /E is dense. So, if we simply stipulate that the compact spaces on which G acts "definably" are simply such quotients G * /E, then by definition there is a universal such action, namely the action of G on S G (M). The notion of "definable amenability" of G from the papers [6], [7] for example, then says precisely that there is a G-invariant Borel probability measure on S G (M). And one could likewise define G to be extremely amenable if there is a G-invariant point in S G (M), namely an invariant type. Now our original motivation was to give an intrinsic description of the actions described in the last paragraph. For a number of reasons, including that of developing a robust theory in analogy with the topological case, we will only do this under a definability of types assumption: all types in S G (M) are definable. As remarked in the introduction important unstable structures such as the field R of real numbers, or the field Q p of p-adic numbers, do have the feature that all types over them are definable. So our theory will apply to real and p-adic semialgebraic Lie groups. Given an NIP theory T , an arbitrary model M of T and group G definable in M, it would be natural to apply the theory we develop to G as a group definable in the expansion M ext . In fact the third author already did this successfully for f sg groups in [13] ASSUMPTION. M is a first order structure, G is a group definable in M, and all types in S G (M) are definable.
So note we do not assume that ALL complete types over M are definable, just complete types extending the formula x ∈ G.
We now give a somewhat strong definition of a definable action of G on a compact space X, suitable for our purposes, under the ASSUMPTION above.
Definition 3.5. Let X be a compact space.
(i) By a definable action of G on X (or definable G-flow) we mean that G acts on X by homeomorphisms, and for each x ∈ X, the map f x : G → X taking g to g · x is definable.
Remark 3.6. (i) The second clause in Definition 3.5(i) is equivalent to saying that the induced map from G to the compact space X X is definable.
(ii) Itai Ben Yaacov suggested to the third author that the topologically correct definition in (i) is that in addition to G acting by homeomorphisms, the map from G to the space Homeo(X) of homeomorphisms of X, where Homeo(X) is equipped with the compact-open topology, is definable when considered as a map from G into the Roelcke compactification of Homeo(X). Under our definability of types assumption, definable G-ambits will have this property. (ii) Moreover for any bounded type-definable over M, G-invariant equivalence relation E on G * , the action of G on G * /E (equipped with the logic topology) is definable.
Proof. (i) Clearly G acts on S G (M) by homeomorphisms. As S G (M) is totally disconnected, the second clause in Definition 3.5(i), says that for any L M -formula ϕ(x), and p(x) ∈ S G (M), {g ∈ G : ϕ(x) ∈ gp} is a definable subset of G. And this follows by definability of p(x).
(ii) This follows from (i): Let X = G * /E and π the (continuous) G-invariant surjection from S G (M) to X. Suppose a ∈ X and C 1 , C 2 are disjoint closed subsets of X. Let D i = π −1 (C i ) for i = 1, 2. So D 1 , D 2 are disjoint closed subsets of S G (M), and by compactness (E being type-definable over M), there are M-definable sets D ′ 1 , D ′ 2 , containing D 1 , D 2 respectively, such that π(D ′ 1 ) and π(D ′ 2 ) are disjoint. Let p(x) ∈ π −1 (a). So as in (i) {g ∈ G : gp ∈ D ′ 1 } is a definable subset Y of G. We see that Y separates {g ∈ G : g · a ∈ C 1 } and {g ∈ G : g · a ∈ C 2 }, as required. Proof. Let (X, G, x 0 ) be a definable G-ambit. Let f : G → X be f (g) = g · x 0 . By Lemma 3.2(i), f extends uniquely to an M-definable map f * from G * to X, and thus a continuous map h : S G (M) → X. Note that h is surjective as its image is compact and contains G · x 0 . Also note that h(1) = x 0 . It remains to see that h is a map of G-spaces. This follows from the construction of f * in the proof of 3.2: for p ∈ S G (M), and g ∈ G, gh(p) = g ϕ∈p f (ϕ(M)) = ϕ∈p gf (ϕ(M)) = ϕ∈p f (gϕ(M)) = ψ∈gp f (ψ(M)) = h(gp) (using the fact that f (ϕ(M)) = ϕ(M) · x 0 ).
We now discuss the semigroup structure on S G (M) So note that if p = g is in G (namely a realized type), then p * q is just gq, so the operation * : S G (M) × S G (M) → S G (M), extends the group action G × X → X. (ii) Given q ∈ S G (M), let r q be the unique continuous function from S G (M) to itself which takes 1 to q (given by Proposition 3.8). Then for any p ∈ S G (M), r q (p) is precisely p * q.
Proof. Left to the reader.
Definition 3.11. (i) By a minimal definable G-flow we mean a definable G-flow (X, G) with no proper closed G-invariant subset (i.e. with no proper G-subflow).
(ii) By a universal minimal definable G-flow, we mean a minimal definable G-flow (I, G) such that for any minimal definable G-flow (X, G) there is a G-map from I to X. (So in fact for every definable G-flow X, minimal or not, there is a G-map from I to X).
Note first that any minimal G-subflow I of S G (M) will be a universal minimal definable G-flow. (Because for any definable G-flow (X, G) there is by Proposition 3.8 a G-map from S G (M) to X whose restriction to I is a G map from I to X. ) On the other hand, there is no a priori reason for there to be a unique universal minimal definable G-flow (in the obvious sense). But in fact uniqueness of the the universal minimal definable G-flow is true. This makes use of the semigroup structure on S G (M), and the proof is simply a copy of any of the proofs in the classical situation. We go briefly through the steps, following Section 3 of [19]. Let I be a minimal subflow of S G (M): Claim. (i) Left ideals (with respect to * ) of S G (M) coincide with closed G-subflows.
(ii) I contains an idempotent p 0 and q * p 0 = q for all q ∈ I.
(iii) Every G-map f : I → I has the form p → p * q for some q ∈ I.

Proof.
Consider h = f • r p 0 : S G (M) → I. Then h = r t where h(1) = t ∈ I. As r p 0 |I is the identity (by (ii)), f and h = r t have the same restriction to I.
Claim. (iv) Every G-map f : I → I is a bijection.
Proof. By (iii) f = r t |I for some t ∈ I. Now I * t being an ideal contained in I coincides with I (by minimality of I) hence s * t = p 0 for some s ∈ I. Let g : I → I be the map r s |I. Then f • g = id. As f was arbitrary all G-maps from I to I are surjective. In particular g is hence f is a bijection.
We conclude from (iv): There is a unique (up to isomorphism of G-spaces) universal minimal definable G-flow, and it coincides with some (any) minimal G-subflow of S G (M).
Proof. Let I be a given minimal subflow of S G (M), and J any other universal minimal G-flow. There are G-maps f : I → J and g : J → I, both necessarily surjective. The composition g • f : I → I is bijective by (iv) above, whereby f is injective, hence a homeomorphism.
So part of the content of [4] was to describe the universal minimal definable G-flow when G = SL(2, R) and M = (R, +, ·).
Finally we discuss amenability. Our definability of types assumption remains in place.
there is a G-invariant Borel probability measure on X.
(ii) G is definably extremely amenable if for every definable G-flow (X, G), X has a fixed point.  [2] as well as [13]. We are not sure about (ii), even in the o-minimal case.
Finally we return to our definability of types assumption on S G (M), and point out that Pestov's characterization ( [10], Theorem 8.1) of extreme amenability passes suitably to the definable category. We say that a definable subset Y of G is left generic if finitely many left translates of Y cover G.

Proposition 3.17. G is (definably) extremely amenable if and only if for each de-
Proof. Suppose first that there is a G-invariant type p ∈ S G (M). We will prove the right hand side without assuming any definability of types assumption. Let X be a left generic definable subset of G. So for some g ∈ G, gX ∈ p. By G-invariance of p, X ∈ p. As p is G-invariant, XX −1 = G. Now for the converse. Given a definable subset Y of G we feel free to identify Y with the clopen subset of S G (M) (and of any closed subset of S G (M)) it defines. So for p ∈ S G (M) we may write p ∈ Y , although this corresponds to the formula defining Y being in the type p. Assume the right hand side. Let M be a minimal closed G-invariant subset of S G (M). We will prove that M is a singleton, by showing that for any p ∈ M, and g ∈ G, gp = p. Suppose for a contradiction that for some g ∈ G and p ∈ S G (M), gp = p. Then there is a formula (i.e. definable subset of G) Y ∈ p such that Y ∩ g(Y ) = ∅. Note that the union U of all gY must cover M, Because if not then M \ U is closed G-invariant and nonempty, contradicting minimality of M. By compactness of M, it is covered by g 1 Y ∪ . . . ∪ g k Y for some g 1 , . . . , g k ∈ G. Let Y 1 = {h ∈ G : Y ∈ h(p)}. By definability of p, Y 1 is a definable subset of G. Claim (I). g 1 Y 1 ∪ . . . ∪ g k Y 1 = G (so Y 1 is left generic in G).

Proof of Claim (I).
Let h ∈ G. So hp ∈ M so hp ∈ g i Y for some i = 1, . . . , k. Hence g −1 i h ∈ Y 1 , and so h ∈ g i Y 1 .

Proof of Claim (II).
If not, let h ∈ Y 1 such that gh ∈ Y 1 . So hp ∈ Y and ghp ∈ Y hence hp ∈ Y ∩ g −1 Y implying that gY ∩ Y = ∅, contradicting choice of Y .
Claims (I) and (II) contradict the right hand side, completing the proof.
Remark 3.18. (i) The proof above (which was adapted from Pestov's proof of Theorem 7.1 in [10]) generalizes to show that the kernel of the action of G on its minimal subflow is precisely the intersection of the sets Y Y −1 for Y ⊆ G definable and left generic.
(ii) If T h(M) is stable, then G is definably extremely amenable iff G is connected, in which case there is a unique invariant type.
(iii) One might try to generalize Proposition 3.17 to the general situation with no definability of types assumption. As an example the following holds: Every minimal subflow of S G (M) is a singleton if and only if whenever Y ⊆ G is definable and (as a clopen subset of S G (M)) meets some minimal subflow of S G (M) then Y Y −1 = G.