SMOOTHNESS OF BOUNDED INVARIANT EQUIVALENCE RELATIONS

KRZYSZTOF KRUPIŃSKI AND TOMASZ RZEPECKI

Abstract. We generalise the main theorems from the paper “The Borel cardinality of Lascar strong types” by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to get the conclusion.

1. Introduction

1.1. Preface. This paper will concern the Borel cardinalities of bounded, invariant equivalence relations, as well as some weak analogues in an uncountable case. More precisely, we are concerned with the connection between type-definability and smoothness of these relations – type-definable equivalence relations are always smooth (cf. Fact 2.7); the converse is not true in general. We also apply this to the study of connected components in definable group extensions.

The general motivation for the use of Borel cardinality in the context of bounded invariant equivalence relations is a better understanding of “spaces” of strong types (i.e., “spaces” of classes of such relations). For a bounded type-definable equivalence relation, its set of classes equipped with the so-called logic topology forms a compact Hausdorff topological space. However, for relations which are only invariant but not type-definable, the logic topology is not necessarily Hausdorff, so it is not so useful. The question arises how to measure the complexity of the spaces of classes of such relations. One of the ideas is to investigate Borel cardinalities of such relations, which was formalised in [KPS13], wherein the authors asked whether the Lascar strong type must be non-smooth if it is not equal to the Kim-Pillay strong type. This question was answered in the positive in [KPS13], and in this paper, we generalise its methods to a more general class of invariant equivalence relations, and we find an important application in the context of definable group extensions.

There are four main results:

(1) Theorem 3.2, a technical statement showing that some invariant equivalence relations are not smooth, which is proved by a simple modification of the proof of the main result of [KMS13] (Very similar results have been since shown in [KM13] using different – though not unrelated – methods, although it should be noted that the latter preprint was circulated after the proof of Theorem 3.2 presented here was found by the authors.); (2) Theorem 3.10 (which is an uncountable analogue of Theorem 3.2) and its Corollary 3.11; again, it is obtained by a modification of a result of [KMS13], although in this case it is somewhat more elaborate; this theorem contains some completely new information concerning the notion of τ topology introduced in this paper, which is essential in the application to definable group extensions in the last section of the paper;

2010 Mathematics Subject Classification. 03C45; 03E15; 03C60.

Key words and phrases. bounded invariant equivalence relations, Borel cardinality, modeltheoretic connected components.

The first author is supported by NCN grant 2012/07/B/ST1/03513.

The paper is based on the Master’s thesis of the second author.

(3) Theorem 4.8, in which we attempt to analyse in detail the connection between smoothness, type-definability and some other properties of bounded and invariant equivalence relations, under some additional assumptions; it uses a corollary of Theorem 3.2 to show that some of these properties are stronger than others, and several (original) examples to show that they are not equivalent;

(4) Theorem 5.2, which applies Corollary 3.11 along with some ideas from [GK13] and [KPS13] in the context of definable group extensions, in order to give a criterion for type-definability of subgroups of such extensions, resulting in a proof of important technical conjectures (see Conjectures 3 and 4 in the last section) from [GK13] in Corollary 5.7; the motivation for these conjectures is recalled in the remark following them.

The main results discussed above are included in Sections 3, 4 and 5. The second section develops the necessary framework upon which we will base the part that comes after it – the language in which we express the sequel. In particular, we introduce the notions of orbital and orbital on types equivalence relations, as well as the notion of a normal form.

1.2. Conventions. In the following, unless otherwise stated, we assume that we have a fixed complete theory T with infinite models. (The theory may be multisorted, and it will, of course, vary in some specific examples.)

We also fix a monster model ℭ ⊧ T, that is, a model which is κ-saturated and strongly κ-homogeneous for κ a sufficiently large cardinal (and whenever we say “small” or “bounded”, we mean smaller than this κ). If we assume that there is a sufficiently large and strongly inaccessible cardinal κ, we can take for ℭ the saturated model of cardinality κ. We say that an equivalence relation on a product of sorts of ℭ is bounded if its number of classes is bounded.

We assume that all parameter sets are contained in ℭ, every model we consider is an elementary substructure of ℭ, and every tuple is of small length. Often, we will denote by M an arbitrary, but fixed small model.

For a small set A ⊆ ℭ, by A-invariant we mean Aut(ℭ/A)-invariant.

For simplicity, whenever we mention definable, type-definable or invariant sets, we mean that they are (unless otherwise stated) ∅-definable, ∅-type-definable or ∅-invariant, respectively.

When talking about tuples of elements of ℭ, we will often say that they are in ℭ (as opposed to some product of various sorts of ℭ), without specifying the length, when it does not matter or there is no risk of confusion. Likewise, we will often write X ⊆ ℭ when X is a subset of some product of sorts of ℭ.

If X is some A-invariant set (esp. type-definable over A), we will denote by S_{X}(A) the set of complete A-types of elements of X, and similarly we will sometimes omit X (or names of sorts in multi-sorted context) in S_{X}(A), and write simply S(A) instead.

Throughout the paper, formulas and types will be routinely identified with the corresponding subsets of ℭ, as well as the corresponding subsets of type spaces (or points, in case of complete types). Similarly, invariant sets will be identified with subsets of type spaces and equivalent L_{∞,ω} formulas. For example, if X ⊆ ℭ is an A-invariant set, then we will identify X with ⋁_{i∈I} ⋀_{j∈J} φ_{i,j}(x, A) (where I, J are possibly infinite index sets and φ_{i,j} are first order formulas) if we have

x ∈ X ⟺ ℭ ⊧ ⋁_{i} ⋀_{j} φ_{i,j}(x, A).

In this case, we also associate with X the subset X_{A} = {tp(a/A) | a ∈ X} of S(A); when A = ∅, and there is no risk of confusion, we will sometimes simply write X instead of X_{∅}.

When metrics are mentioned, they are binary functions into [0, ∞] = R_{≥0} ∪{∞} satisfying the usual axioms (coincidence axiom, symmetry and triangle inequality), but in particular, they are allowed to (and usually will) assume ∞.

1.3. Preliminaries. It is assumed that the reader is familiar with basic concepts of model theory (e.g. compactness, definable sets, type-definable sets, type spaces, saturated models, indiscernible sequences) and descriptive set theory (e.g. Polish spaces, standard Borel spaces, Borel classes).

Furthermore, we will also use some well-known (but less widely known) facts and terms related to the following subjects.

(1) Borel cardinalities of Borel equivalence relations ([Kan08, in particular Chapter 5], [BK96, esp. Chapter 3]). For a concise exposition of fundamental issues concerning this topic, the reader is referred to the preliminary sections in [KPS13] or [KMS13]. Let us only recall here that for Borel equivalence relations E and F on Polish (or, more generally, standard Borel) spaces X and Y, respectively, we say that E is Borel reducible to F, or that the Borel cardinality of E is less than or equal to the Borel cardinality of F (written E ≤_{B} F) if there is a Borel reduction from E to F, i.e. a Borel function f : X → Y such that x_{0} E x_{1} ⟺ f(x_{0}) F f(x_{1}) for all x_{0}, x_{1} ∈ X; the relations E and F are Borel bireducible, or of the same Borel cardinality (written E ∼_{B} F) if E ≤_{B} F and F ≤_{B} E. The relation E is smooth if E ≤_{B} Δ(Z), where Z is a Polish space and Δ(Z) is the equality on Z.

(2) Strong Choquet topological spaces ([Kec95]).

(3) Lascar and Kim-Pillay strong types ([Cas+01, esp. first section]). In this paper, the relation of having the same Lascar strong type (i.e. the finest bounded invariant equivalence relation) will be denoted by ≡_{L}, and the relation of having the same Kim-Pillay strong type (i.e. the finest bounded type-definable equivalence relation) will be denoted by ≡_{KP}.

(4) Model-theoretic connected group components ([GN08] and [Gis11, first two sections]). Recall that for a group G definable in the monster model, G^{000} denotes the smallest invariant subgroup of bounded index, and G^{00} – the smallest type-definable subgroup of bounded index (both considered here without parameters, as explained in the introduction).

(5) The logic topology ([Pil04, Section 2]). Recall that when E is a typedefinable equivalence relation on a type-definable set X, then a subset D ⊆ X/E is closed if and only if its preimage by the quotient map is typedefinable with parameters.

2. Framework

2.1. Bounded invariant equivalence relations. In this chapter, we extend the theory of Borel cardinality of Lascar strong types as considered in [KPS13] to general invariant and bounded equivalence relations, to provide a uniform way of viewing bounded, invariant equivalence relations as relations on topological spaces, which will be standard Borel spaces in the countable case.

Definition. Suppose P is a product of sorts of ℭ. We say that P is countable if it is a product of countably many sorts.

Definition. Suppose X is a subset of some product of sorts P. Then we say that P is the support of X, and we say that X is countably supported if P is countable (according to the preceding definition), and, more generally, say that it is λ-supported for a cardinal λ if P is a λ-fold product.

Definition (Borel invariant set, Borel class of an invariant set). For any invariant set X, we say that X is Borel if the corresponding subset of S(∅) is, and in this case by Borel class of X we mean the Borel class of the corresponding subset of S(∅) (e.g. we say that X is F_{σ} if the corresponding set in S(∅) is F_{σ}, and we might say that X is clopen if the corresponding subset of S(∅) is clopen, i.e. if X is definable).

Similarly if X is A-invariant, we say that it is Borel over A if the corresponding subset of S(A) is (and Borel class is understood analogously).

We say that a set is pseudo-closed if it is closed over some small set (equivalently, if it is type-definable with parameters from a small set).

Remark. Notice that if both the language and A are countable and X is countably supported and Borel over A, then S_{X}(A) – endowed with the σ-algebra generated by formulas over A – is a standard Borel space.

We will use the following descriptive-set-theoretic lemma several times.

Lemma 2.1 ([Kec95, Exercise 24.20]). Suppose X, Y are compact, Polish spaces and f: X → Y is a continuous, surjective map. Then f has a Borel section, so in particular, for any B ⊆ Y, f^{−1}[B] is Borel if and only if B is. Moreover, if they are Borel, then the two are of the same Borel class.

The next corollary says that, in the countable case, when X is invariant over a countable, we need not specify the parameter set in order to talk about the Borel class of X. It is a generalisation of a well-known fact for sets which are definable or type-definable with parameters.

Corollary 2.2. Let A, B be any small sets. Suppose X is an A-invariant and Binvariant subset of a small product of sorts. Then if the support of X, the language, A and B are all countable, then the Borel class of X over A is the same as the Borel class of X over B (in particular, X is Borel over A if and only if it is Borel over B).

Without assumptions of countability, if X is closed or F_{σ} over A, it is also closed or F_{σ} (respectively) over B.

Proof. Without loss of generality, we can assume that A ⊆ B. Then the restriction map f: S(B) → S(A) is a continuous surjection, and f^{−1}[X_{A}] = X_{B}, so by Lemma 2.1, we get the result for the first part.

The second part is true because S(B) → S(A) is continuous, as well as closed (as a continuous map between compact spaces). □

The following definition is somewhat self-explanatory, but since we are going to use it quite often, it should be stated explicitly.

Definition. We say that an invariant equivalence relation E on X refines type if for any a, b ∈ X whenever a E b, then a ≡ b (i.e. tp(a/∅) = tp(b/∅)). Equivalently, E refines type if E ⊆ ≡↾_{X}.

Similarly, we say that E refines Kim-Pillay strong type ≡_{KP} if E ⊆ ≡_{KP} ↾_{X} and likewise we say that Kim-Pillay type refines E if ≡_{KP} ↾_{X} ⊆ E.

The next definition is very important; it will be used to interpret a bounded, invariant equivalence relation E as an abstract equivalence relation on a Polish space. It is a mild generalisation of E_{L}^{M} and E_{KP}^{M} as introduced in [KPS13].

Definition. Suppose E is a bounded, invariant equivalence relation on an invariant set X, while M is a model.

Then we define E^{M} ⊆ S_{X}(M)^{2} ⊆ S(M)^{2} as the relation

p E^{M} q ⟺ there are some a ⊧ p and b ⊧ q such that a E b.

(And the next fact tells us that E-classes are M-invariant, so this is equivalent to saying that for all a ⊧ p, b ⊧ q we have a E b, which implies that E^{M} is an equivalence relation.)

The next fact shows that E^{M} is well-behaved in the sense explained in parentheses, and the Borel classes of E^{M} and E are the same in the countable case (which justifies the definition of Borel class of E at the beginning of this subsection).

Proposition 2.3 (generalisation of [KPS13, Remark 2.2(i)]). Consider a model M, and some bounded, invariant equivalence relation E on an invariant subset X of a product of sorts P.

Consider the natural restriction map π: S_{P^{2}}(M) → S_{P} (M)^{2} (i.e. π(tp(a, b/M)) = (tp(a/M), tp(b/M))). Then we have the following facts:

• Each E-class is M-invariant, in particular

a E b ⟺ tp(a, b/M) ∈ E_{M} ⟺ tp(a/M) E^{M} tp(b/M)

and π^{−1}[E^{M}] = E_{M}.

• If one of E^{M}, E_{M}, E (considered as a subset of S_{P^{2}}(∅)) is closed or F_{σ}, then all of them are closed or F_{σ} (respectively). In the countable case (when the support of E, the language and M are all countable), we have more generally that the Borel classes of E^{M}, E_{M}, E are all the same.

• Similarly – for M-invariant Y ⊆ X – E^{M}↾_{Y_{M}} is closed or F_{σ} [or Borel in the countable case] if and only if E_{M} ∩ (Y^{2})_{M} is.

Proof. For the first bullet, notice that E is refined by (a restriction of) Lascar strong type (cf. [Cas+01, Fact 1.4]), which in turn is refined by equivalence over M (for any model M, cf. [Cas+01, Fact 1.12]), and therefore any points equivalent over M are also Lascar equivalent, and therefore E-equivalent.

The second bullet is similar to Corollary 2.2: it is a consequence of the fact that π and the restriction map S_{P^{2}}(M) → S_{P^{2}}(∅) are both continuous and closed (because S_{P^{2}}(M) is compact). For the countable case, we use Lemma 2.1.

The last part follows analogously, as π^{−1}[E^{M}↾_{Y_{M}}] = E_{M} ∩ (Y^{2})_{M}. □

The next two facts will be used in conjunction with Corollary 2.2 to show that some E-saturated sets (where E is a bounded, invariant equivalence relation) are closed or F_{σ} over any model M.

Corollary 2.4. If E is a bounded, invariant equivalence relation on X and Y ⊆ X is E-saturated (i.e. containing any E-class intersecting it), then for any model M, Y is M-invariant.

Proof. Since Y is E-saturated, it is a union of E-classes, each of which is setwise M-invariant. □

We also have a version for groups.

Corollary 2.5. If G is an invariant group and H is a subgroup of G containing some invariant subgroup of bounded index (equivalently, H contains G^{000}), then every coset of H (including H itself) is invariant over any model M.

Proof. Immediate from the previous corollary with E being the relation of being in the same coset of G^{000}. □

The next proposition establishes a notion of Borel cardinality.

Proposition 2.6 (generalisation of [KPS13, Proposition 2.3]). Assume that the language is countable. For any E which is a bounded (invariant) Borel equivalence relation on some type-definable and countably supported set X, and if Y ⊆ X is pseudo-closed and E-saturated, then the Borel cardinality of the restriction of E^{M} to Y_{M} does not depend on the choice of the countable model M. In particular, if X = Y, the Borel cardinality of E^{M} does not depend on the choice of the countable model M.

Proof. Follows from Lemma 2.1 analogously to [KPS13, Proposition 2.3]. (Note that because of Proposition 2.3 and Corollary 2.4 the relations E^{M} and E^{M}↾_{Y_{M}} are well-defined Borel equivalence relations on Polish spaces.) □

We have thus justified the following definition.

Definition. If E is as in the previous proposition, then by Borel cardinality of E we mean the Borel cardinality of E^{M} for a countable model M. Likewise, we say that E is smooth if E^{M} is smooth for a countable model M.

Similarly, if Y is pseudo-closed and E-saturated, the Borel cardinality of E↾_{Y} is the Borel cardinality of E^{M}↾_{Y_{M}} for a countable model M.

Fact 2.7. A bounded, type-definable equivalence relation is smooth. Similarly, if the restriction of a bounded, invariant equivalence relation to a saturated, pseudo-closed set Y is relatively type-definable, then the restriction is smooth.

Proof. If E is type-definable, then so is its domain, and the corresponding subset of S(M)^{2} is closed (by Proposition 2.3), and in particular G_{δ}, and therefore smooth (cf. for example [BK96, Theorem 3.4.3]). The proof of the second part is analogous: the Borel cardinality of the restriction of E to Y is the Borel cardinality of E^{M} ∩(Y_{M})^{2}, which is closed in (Y_{M})^{2}, and thus smooth. □

2.2. Normal forms. In this subsection, we introduce some more specific kinds of invariant equivalence relations, which naturally arise when we interpret the main result.

Definition (Normal form). If Φ_{n}(x, y) is a sequence of (partial) types on a typedefinable set X such that Φ_{0}(x, y) = ((x = y) ∧ x ∈ X) and which is increasing (i.e. for all n, Φ_{n}(x, y) ⊢ Φ_{n+1}(x, y)), then we say that ⋁_{n∈N} Φ_{n}(x, y) is a normal form for an invariant equivalence relation E on X if we have for any a, b an equivalence a E b ⟺ ℭ ⊧ ⋁_{n∈N} Φ_{n}(a, b), and if the binary function d = d_{Φ} : ℭ^{2} → N ∪ {∞} defined as

d(a, b) = min{n ∈ N | ℭ ⊧ Φ_{n}(a, b)}

(where min ∅ = ∞) is an invariant metric with possibly infinite values – that is, it satisfies the axioms of coincidence, symmetry and triangle inequality. In this case, we say that d induces E on X.

Example 2.8. The prototypical example of a normal form is ⋁_{n} d_{L}(x, y) ≤ n, inducing ≡_{L}, and d_{L} is the associated metric (where ≡_{L} is the relation of having the same Lascar strong type and d_{L} is the Lascar distance).

Remark. The Lascar distance, by its very definition, has the nice property that it is “geodesic” in the sense that if two points a, b are at distance n, then there is a sequence of points a = a_{0}, a_{1}, …, a_{n} = b such that each pair of successive points is at distance 1. The metrics obtained from normal forms usually will not have this property (notice that existence of such a “geodesic” metric for E is equivalent to E being the transitive closure of a type-definable relation).

Example 2.9. If Φ_{n}(x, y) is an increasing sequence of type-definable equivalence relations, then ⋁_{n} Φ_{n}(x, y) is trivially a normal form. In particular, if E = Φ(x, y) is type-definable, then we can put – for all n > 0 – Φ_{n}(x, y) = Φ(x, y), yielding a somewhat degenerate normal form for E.

Definition. If we have an invariant equivalence relation E on a type-definable set X with a normal form ⋁_{n∈N} Φ_{n}(x, y), corresponding to a metric d, and Y ⊆ X is some set, then the diameter of Y is the supremum of d-distances between points in Y.

Fact 2.10. If E is as above, and X is (the set of realisations of) a single complete type, then all E-classes have the same diameter (because the Aut(ℭ) acts transitively on X in this case, and the diameter is invariant under automorphisms).

The following proposition is the essential step in adapting the techniques of [KMS13] to prove Theorem 3.2.

Proposition 2.11. Suppose E is an F_{σ} (over ∅), bounded equivalence relation on a type-definable set X. Then E has a normal form ⋁_{n} Φ_{n} such that Φ_{1}(x, y) holds for any x, y which are terms of an infinite indiscernible sequence. (This implies that for any a, b, if d_{L}(a, b) ≤ n, then ⊧ Φ_{n}(a, b), so that the induced metric satisfies d ≤ d_{L}. It also shows that every F_{σ} equivalence relation has a normal form.)

Proof. As E is bounded, the Lascar strong type restricted to X is a refinement of E (cf. [Cas+01, Fact 1.4]), and hence E ∪ (≡_{L}↾_{X}) = E. In addition, since E is F_{σ}, we can find types Φ_{n}(x, y) such that x E y ⟺ ℭ ⊧ ⋁_{n} Φ_{n}(x, y).

Consider the sequence Φ′_{n}(x, y) of types, defined recursively by:

(1) Φ′_{0}(x, y) = ((x = y) ∧ x ∈ X),

(2) Φ′_{1}(x, y) = (Φ_{1}(x, y) ∨ Φ_{1}(y, x) ∨ x = y ∨ d_{L}(x, y) ≤ 1) ∧ (x, y ∈ X),

(3) Φ′_{n+1}(x, y) = Φ_{n+1}(x, y) ∨ Φ_{n+1}(y, x) ∨ (∃z)(Φ′_{n}(x, z) ∧ Φ′_{n}(z, y)). It is easy to see that ⋁ Φ′_{n} is a normal form and represents the smallest equivalence relation containing E and ≡_{L}↾_{X} (as a set of pairs), which is just E, and d_{L}(x, y) ≤ 1 (i.e. the statement that x, y are in an infinite indiscernible sequence) implies Φ′_{1}(x, y) by the definition.

The statement in the parentheses follows from the fact that d_{L}(a, b) ≤ n is defined as the n-fold composition of d_{L}(a, b) ≤ 1. □

The theorem of Newelski we will see shortly is a motivating example for the study of Borel cardinality: it can be interpreted as saying that some equivalence relations have Borel cardinality of at least Δ(2^{N}). We will see later in Corollary 3.3 that for E which are orbital (a concept which we will define soon), we can strengthen this result to replace Δ(2^{N}) with E_{0}, and this is optimal in the sense explained in a remark after Corollary 3.3.

Theorem 2.12 ([Corollary 1.12][New03]). Assume x E y is an equivalence relation refining ≡, with normal form ⋁_{n∈N} Φ_{n}. Assume p ∈ S(∅) and Y ⊆ p(ℭ) is pseudoclosed and E-saturated. Then either E is equivalent on Y to some Φ_{n}(x, y) (and therefore E is relatively type-definable on Y), or |Y/E| ≥ 2^{ℵ_{0}}.

Remark. Newelski uses a slightly more stringent definition of a normal form (which we may enforce in all interesting cases without any significant loss of generality), i.e. that d satisfies not only triangle inequality, but also

d(a, b), d(b, c) ≤ n ⟹ d(a, c) ≤ n + 1.

The definition used in this paper is sufficient for the previous theorem, and in addition, it has the added benefit of being satisfied by the Lascar distance d_{L}, and it seems more natural in general.

The following corollary allows us some freedom with regards to the normal form, allowing us to replace – in some cases – any normal form with one chosen as in Proposition 2.11, without loss of generality.

Corollary 2.13. Suppose E is an F_{σ} equivalence relation on a type-definable set, and that E refines ≡. Then for any class C of E, the following are equivalent:

(1) C is pseudo-closed,

(2) C has finite diameter with respect to each normal form of E (i.e. it has finite diameter with respect to the metric induced by each normal form),

(3) C has finite diameter with respect to some normal form of E. In addition, if E is bounded and all E-classes satisfy these conditions, then E is refined by ≡_{KP} (restricted to its domain).

Proof. Assume that C is pseudo-closed. Setting Y = C in Theorem 2.12, we immediately get that C has finite diameter with respect to any normal form of E. Implication from the second condition to third follows from the fact that E has a normal form by the previous proposition, and the implication from third to first is trivial.

“In addition” can be obtained as follows. E refines ≡, so it is enough to show that the restriction of E to any p ∈ S(∅) is refined by the restriction of ≡_{KP} to p. But any class in the restriction has finite diameter with respect to some normal form, and they all have the same diameter (by Fact 2.10), so in fact, the restriction is type-definable and as such refined by ≡_{KP} (cf. [Cas+01, Fact 1.4]). □

Example 2.14. The above is no longer true if we allow E to be refined by ≡. For example, consider the theory T = Th(R, +, ·, 0, 1, <) of real closed fields, and the total relation on the entire model. Clearly, it has a normal form {x = y}∨⋁_{n>0}(x = x), and the induced metric is just the discrete 0-1 metric, and in particular its only class (the entire model) has diameter 1. On the other hand, we might give it a normal form {x = y} ∨ ⋁_{n>0}(⋀_{m≥n}(x = m ↔ y = m)) (where m ranges over natural numbers). With respect to this normal form, any two distinct positive natural numbers k, l are at distance max(k, l) + 1. In particular, the diameter of the only class is infinite.

Remark. If E is a type-definable equivalence relation, then its classes are trivially pseudo-closed, so by Corollary 2.13, if E is refined by ≡, then for any normal form of E, all E-classes have finite diameter.

2.3. Orbital equivalence relations. For technical reasons, later on we will rely on the action of a group of automorphisms, so we introduce the following definition.

Definition (Orbital equivalence relation, orbital on types equivalence relation). Suppose E is an invariant equivalence relation on a set X.

• We say that E is orbital if there is a group Γ ≤ Aut(ℭ) such that Γ preserves classes of E setwise and acts transitively on each class.

• We say that E is orbital on types if it refines type and the restriction of E to any complete ∅-type is orbital.

Remarks.

• An orbital equivalence relation always refines type. (So every orbital equivalence relation is orbital on types.)

• The relations ≡_{L}, ≡_{KP} are orbital (as witnessed by Aut f_{L}(ℭ), Aut f_{KP} (ℭ)).

• The group witnessing that a given relation is orbital can always be chosen as a normal subgroup of Aut(ℭ) (as we can replace it with its normal closure).

The following proposition shows that the definition of an orbital on types equivalence relation is, in a way, the weakest possible for the proof of Theorem 3.2.

Proposition 2.15. An invariant equivalence relation E refining type is orbital on types if and only if for any class C of E there is a group of automorphisms Γ which preserves E classes within the (complete ∅-)type p containing C, and acts transitively on C.

Proof. The implication (⇒) is clearly a weakening. For (⇐), observe that Aut(ℭ) acts transitively on X := p(ℭ), so for any class C′ ∈ X/E we have an automorphism σ which takes C to C′. It is easy to see that then σΓσ^{−1} acts transitively on C′ and preserves all E-classes in X setwise. From that we conclude that the normal closure of Γ in Aut(ℭ) witnesses that E restricted to X is orbital. □

The following simple corollary allows us to easily recognise some relations as orbital on types.

Corollary 2.16. If E is an invariant equivalence relation on an invariant set X, refining ≡, and the restriction of E to any complete type in X has at most two classes, then E is orbital on types.

Proof. Without loss of generality we may assume that X is a single complete type, so Aut(ℭ) acts transitively on X. In particular, for any element a ∈ X, we have a set S ⊆ Aut(ℭ) such that S · a = [a]_{E}. Since E is invariant, elements of S preserve [a]_{E} and so does the group Γ = ⟨S⟩.

Of course, Γ preserves X, so it also preserves the complement X \ [a]_{E}. But since E has at most two classes, this means that Γ preserves all classes, so by the previous proposition, E is orbital on types. □

At a glance, it is not obvious whether the condition that E is orbital on types is any stronger than the condition that it refines type. The following examples show that it is indeed the case.

Example 2.17. Consider the permutation group

G = ⟨(1, 2)(3, 5)(4, 6), (1, 3, 6)(2, 4, 5)⟩
= {(), (1, 2)(3, 5)(4, 6), (1, 3, 6)(2, 4, 5),
(1, 4)(2, 3)(5, 6), (1, 5)(2, 6)(3, 4), (1, 6, 3)(2, 5, 4)}

acting naturally on a 6-element set. Then the equivalence relation ∼ such that 1 ∼ 2, 3 ∼ 4, 5 ∼ 6 (and no other nontrivial relations) is preserved by G, but it is not the orbital equivalence relation of any subgroup (in fact, the subgroup of G preserving all ∼-classes setwise is trivial).

Let M_{0} be a structure with base set {1, 2, 3, 4, 5, 6}, with a relation symbol E for ∼, and such that G is the automorphism group of M_{0} (which we can obtain, for instance, by adding a predicate for the set of all orbits of G on M_{0}^{6}).

Then E is an invariant (even definable) equivalence relation which refines ≡ and is not orbital on types.

We can extend Example 2.17 to an infinite model in a number of simple ways, for instance, by taking a product with an infinite trivial structure.

We finish with a little less artificial example.

Example 2.18. Consider a large algebraically closed field K of characteristic p > 0, and choose some t ∈ K, transcendental over the prime field F_{p}, and consider T = Th(K, +, ·, t).

Let n > 3 be a natural number which is not divisible p, and X be the set of n-th roots of t in K (i.e. the roots of x^{n} − t). Notice that X generates a definable, finite additive group ⟨X⟩. Let us introduce

G = ({a = (a_{1}, a_{2}) ∈ K^{2} | a_{1} + a_{2} ∈ ⟨X⟩}, +).

G is a definable group (definably isomorphic to K ×⟨X⟩). Consider the equivalence relation on G defined by

a E b ⟺ (a ≡ b ∧ a_{1} + a_{2} = b_{1} + b_{2}).

We will show that E is not orbital on types, even though it is type-definable, bounded and refines ≡. (N.b. this E is the conjunction of ≡ and the relation of lying in the same coset of G^{000}, which in this case is equal to G^{0}.)

Let ξ be some primitive nth root of unity. One can easily check that for any x_{1}, x_{2} ∈ X, the pairs (x_{1}, ξ) and (x_{2}, ξ^{−1}) have the same type, which implies that all a ∈ G of the form (x, ξ^{±1}x), where x ∈ X, have the same type, say p_{0} ∈ S_{G}(∅). For any x ∈ X we also have (x, ξx) E (ξx, x). Thus, if E was orbital on types, there would be some automorphism f ∈ Aut(K/t) which takes x to ξx and ξx to x – therefore taking ξ to ξ^{−1} – which preserves setwise the E-classes within p_{0}. But then

b = f((ξx, ξ^{2}x)) = (x, ξ^{−1}x) ¬E (ξx, ξ^{2}x) = a ⊧ p_{0},

because a_{1} + a_{2} − b_{1} − b_{2} = x(ξ + ξ^{2} − 1 − ξ^{−1}) = ξ^{−1}x(ξ^{3} + ξ^{2} − ξ^{1} − 1) and ξ is algebraic of degree n > 3.

We have seen that the E-class of (ξx, ξ^{2}x) ⊧ p_{0} is not preserved by f, a contradiction.

2.4. Invariant subgroups as invariant equivalence relations. We start from the following obvious definition.

Definition. Suppose G is a type-definable group and H ≤ G is invariant. We define E_{H} as the relation on G of lying in the same right coset of H.

Remark. Clearly, E_{H} is invariant, and it has [G : H] classes, so H has bounded index if and only if E_{H} is a bounded equivalence relation.

It is not hard to see that invariant subgroups of type-definable groups correspond to invariant equivalence relations as shown in the following proposition.

Lemma 2.19. Suppose G is a type-definable group and H ≤ G is an invariant subgroup. Then E_{H} is type-definable or F_{σ} if and only if H is type-definable or F_{σ}, respectively.

Proof. Consider the mapping f: S_{G^{2}}(∅) → S_{G}(∅) given by tp(a, b/∅) ↦ tp(ab^{−1}/∅). Since the operations in G are type-definable, this map is a well-defined, continuous and closed (by compactness) surjection, and E_{H} = f^{−1}[H]. □

Remarks.

• The previous proposition would remain true if we had taken for E_{H} the relation of lying in the same left coset, but right cosets will be technically more convenient in a short while.

• Equivalence relations E_{H} do not refine type, and in particular are not orbital on types, which will be needed later on. We will resolve this issue shortly by choosing a different equivalence relation to represent H, which will be closely related to E_{H} (in a way, homeomorphically equivalent) and orbital on types for normal H.

The theorem below will allow us to “transform” the relation E_{H} to an equivalence relation on a single type.

Theorem 2.20 (see [GN08, Section 3, in particular Propositions 3.3 and 3.4]). If G is a definable group, and we adjoin to ℭ a left principal homogeneous space 𝔛 of G (as a new sort; we might think of it as an “affine copy of G”), along with a binary function symbol for the left action of G on 𝔛, then the Kim-Pillay and Lascar strong types correspond exactly to the orbit equivalence relations of G^{00} and G^{000} acting on 𝔛. Moreover, we have isomorphisms:

Aut((ℭ, 𝔛, ·)) ≅ G ⋊ Aut(ℭ),
Aut f_{KP} ((ℭ, 𝔛, ·)) ≅ G^{00} ⋊ Aut f_{KP} (ℭ),
Aut f_{L}((ℭ, 𝔛, ·)) ≅ G^{000} ⋊ Aut f_{L}(ℭ).

Where:

(1) the semidirect product is induced by the natural action of Aut(ℭ) on G,

(2) on ℭ, the action of Aut(ℭ) is natural, and that of G is trivial,

(3) on 𝔛 we define the action by fixing some x_{0} and putting – for g ∈ G and σ ∈ Aut(ℭ) – σ_{g}(h · x_{0}) = (hg^{−1})x_{0} and σ(h · x_{0}) = σ(h) · x_{0}.

Remark. The isomorphisms are not canonical in general: they depend on the choice of the base point x_{0}.

Until the end of this subsection, we fix a definable group G and the structure (ℭ, 𝔛, ·) as above. Note that a definable group is always finitely (and therefore countably) supported.

Definition. Let H be an invariant subgroup of G. Then E_{H,X} is the relation on 𝔛 of being in the same H-orbit.

Proposition 2.21. The mapping Φ: H ↦ E_{H,X} is a bijection between invariant subgroups of G and invariant equivalence relations on 𝔛.

Proof. We fix some x_{0} ∈ 𝔛, so as to apply the description of the automorphism group of (ℭ, 𝔛, ·) from Theorem 2.20.

First, choose some invariant H ≤ G. We will show that E_{H,X} is invariant. By the definition of E_{H,X} and Theorem 2.20, it is enough to show that for arbitrary h ∈ H, σ ∈ Aut(ℭ) and g, k ∈ G, one has σ(kx_{0}) E_{H,X} σ(hkx_{0}) and kgx_{0} E_{H,X} hkgx_{0}. The latter is immediate by the definition of E_{H,X}. For the former, just see that

σ(kx_{0}) = σ(k)x_{0} E_{H,X} σ(h)σ(k)x_{0} = σ(hkx_{0}),

because σ(h) ∈ H (by invariance of H).

To see that Φ is a bijection, choose an arbitrary invariant equivalence relation E on 𝔛, and let H be the setwise stabiliser of [x_{0}]_{E}. Take arbitrary h ∈ H, σ ∈ Aut(ℭ). Then

x_{0} E hx_{0} ⟹ x_{0} = σ(x_{0}) E σ(hx_{0}) = σ(h)x_{0},

therefore σ(h) ∈ H, and since h and σ were arbitrary, H is invariant. To see that E = E_{H,X}, notice that for any x_{1} = k_{1}x_{0} and x_{2} = k_{2}x_{0} we have

k_{1}x_{0} E k_{2}x_{0} ⟺ x_{0} E k_{2}k_{1}^{−1}x_{0} ⟺ k_{2}k_{1}^{−1} ∈ H ⟺ (∃h ∈ H) hk_{1}x_{0} = k_{2}x_{0}. □

Remark. An invariant subgroup H ≤ G has bounded index if and only if E_{H,X} is a bounded equivalence relation.

Proposition 2.22. Let H ≤ G be an invariant subgroup of bounded index and let K be a pseudo-closed subgroup such that H ≤ K ≤ G.

Let M ⪯ ℭ be any small model. Then, if we put N = (M, G(M) · x_{0}) ⪯ (ℭ, 𝔛, ·), the map g ↦ g · x_{0} induces a homeomorphism S_{G}(M) → S_{X}(N) which takes E_{H}^{M} to E_{H,X}^{N} and K_{M} to (K · x_{0})_{N}.

In particular:

• E_{H,X} is closed or F_{σ} if and only if E_{H} is (respectively),

• if the language is countable, while H is F_{σ} (or even Borel), then the Borel cardinalities of E_{H}↾_{K} and E_{H,X}↾_{K·x_{0}} coincide.

Proof. The map f: S_{G}(N) → S_{X}(N) defined by f(tp(g/N)) = tp(g · x_{0}/N) is a homeomorphism (because it is induced by an N-definable bijection), and f takes K_{N} to (K · x_{0})_{N} and E_{H}^{N} to E_{H,X}^{N}. It is also easy to see that the restriction map g: S_{G}(N) → S_{G}(M) (with the latter considered in the original structure ℭ) is also a homeomorphism, which takes E_{H}^{N} to E_{H}^{M} and K_{N} to K_{M}. The rest is now clear. □

Using this language, we have the following corollary of Theorem 2.12:

Corollary 2.23. Suppose G is a definable group and H ≤ G is a type-definable subgroup. Suppose in addition that H = ⋃ C_{n}, where C_{n} are type-definable, symmetric sets containing e and such that C_{n}^{2} ⊆ C_{n+1}. Then for some n we have H = C_{n}.

Proof. Consider the equivalence relation E_{H,X}. Then C_{n,X} := {(x, x′) ∈ 𝔛^{2} | x ∈ C_{n}x′} give us a normal form for this relation, which is type-definable and only defined on a single type, so the result follows from Corollary 2.13. □

We finish with an observation that allows us to easily see that some E_{H,X} are orbital.

Proposition 2.24. Suppose H is a normal, invariant subgroup of G. Then E_{H,X} is orbital as witnessed by H ≤ Aut((ℭ, 𝔛, ·)).

Proof. Consider the action ∗ of H on (ℭ, 𝔛, ·) by automorphisms. Then – because H is a normal subgroup of G – we have for any x = g · x_{0} ∈ 𝔛 that

H ∗ (g · x_{0}) = (gH^{−1}) · x_{0} = (gH) · x_{0} = (Hg) · x_{0} = H · (g · x_{0}) = [x]_{E_{H,X}} ,

and hence H ≤ G ⋊ Aut(ℭ) = Aut((ℭ, 𝔛, ·)) witnesses that E_{H,X} is orbital. □

Remark. The converse of the previous proposition is not true: if we have G = S_{3}, H = ⟨(1, 2)⟩ and Aut(ℭ) acting on G in such a way that any σ ∈ Aut(ℭ) acts on G either trivially or by conjugation by (1, 2), then although H is not normal, E_{H,X} is orbital: for σ ∈ Aut(ℭ) acting nontrivially on G we have

((1, 2), σ)(g · x_{0}) = ((1, 2) · g · (1, 2)^{−1}) · (1, 2)^{−1} · x_{0} = (1, 2) · (g · x_{0}).

3. The technical theorem

3.1. The countable case. As before, when E is an invariant, bounded equivalence relation, we denote by E^{M} the induced equivalence relation on S(M). For the statement of the next corollary, we need to extend the notion of distance to the type spaces.

Definition. If E is an F_{σ} equivalence relation induced by a metric d (coming from some normal form), then we also denote by d_{M} the induced distance on S(M), i.e.

d_{M}(p_{1}, p_{2}) = min_{a_{1}⊧p_{1},a_{2}⊧p_{2}} d(a_{1}, a_{2}).

Remark. The classes of E^{M} are precisely the “metric components” of d_{M}, i.e. the maximal sets of types which are pairwise in finite distance of one another in the sense of d_{M}, though d_{M} might not satisfy the triangle inequality, so it is not in general a metric.

We will use the next theorem to show Theorem 3.2.

Theorem 3.1 (based on [KMS13, Corollary 2.3]). Suppose we have:

• a countable theory T with monster model ℭ,

• a countable model M ⪯ ℭ,

• a type-definable, countably supported set X,

• a bounded F_{σ} equivalence relation E on X, with normal form ⋁_{n} Φ_{n}, inducing metric d,

• a pseudo-closed and E-saturated Y ⊆ X. Assume in addition that there is some p ∈ Y_{M} ⊆ S_{X}(M) such that for every formula φ ∈ p with parameters in M, and for all N ∈ N, there is some σ ∈ Aut(ℭ) such that:

(1) σ fixes M and all E-classes in Y setwise (and therefore Y itself as well),

(2) φ ∈ σ(p) and N < d_{M}(σ(p), p). Then there is a continuous, injective homomorphism

(2^{N}, E_{0}, ¬E_{0}) → (Y_{M}, E^{M}↾_{Y_{M}} , ¬(E^{M}↾_{Y_{M}})).

In particular, E^{M}↾_{Y_{M}} is not smooth.

Proof. The proof is the same as that of [KMS13, Corollary 2.3]. The only difference is that for Γ we take the group of automorphisms of ℭ which fix M and all E-classes in Y setwise (instead of all Lascar strong types as there), and we use d_{M} instead of the Lascar distance. Note that Y_{M} is Polish by Corollary 2.2 and Corollary 2.4. □

The above implies the next theorem. As mentioned in the introduction, a similar theorem has been proved, independently, in [KM13] using different methods. The proof we list here is a generalization of the main result of [KMS13], where the relation in question is the Lascar strong type.

Theorem 3.2 (based on [KMS13, Theorem 4.12]). We are working in the monster model ℭ of a complete, countable theory. Suppose we have:

• a type-definable, countably supported set X,

• a bounded, F_{σ} equivalence relation E on X, which is orbital on types,

• a pseudo-closed and E-saturated set Y ⊆ X,

• an E-class C ⊆ Y with infinite diameter with respect to some normal form

of E, Then E↾_{Y} is not smooth.

Proof. By Proposition 2.11 and Corollary 2.13, we can choose a normal form for E such that the induced distance d satisfies d ≤ d_{L}, with respect to which C has infinite diameter. We can also assume that X is the complete type containing C (restricting Y to this type), so that E is orbital as witnessed by some group Γ.

Then we proceed as in Theorem 4.12 of [KMS13] (aiming to use Theorem 3.1), only instead of Aut f_{L}(ℭ) we use Γ (note that all the facts about generic and proper types and formulas from [KMS13] still hold with Γ replacing Aut f_{L}(ℭ), because Γ acts transitively on C), and instead of Lascar distance we use d. □

Remark. We can always take for Γ the group of all automorphisms preserving Eclasses setwise. (In which case Γ ⊴ Aut(ℭ).)

The next corollary can be seen as a strengthening of Theorem 2.12 in case of E which are orbital on types (because a relation with countably many classes is smooth).

Corollary 3.3. Assume the language is countable. Suppose E is a bounded, F_{σ} and orbital on types equivalence relation on a type-definable and countably supported set X. Let a ∈ X be arbitrary, and assume that Y ⊆ [a]_{≡} is E-saturated, pseudo-closed with a ∈ Y. Fix any normal form ⋁_{n} Φ_{n} for E. Then the following are equivalent:

(1) E↾_{Y} is smooth,

(2) E↾_{[a]_{≡}} is type-definable,

(3) all E-classes in [a]_{≡} have finite diameter with respect to ⋁_{n} Φ_{n},

(4) all E-classes in [a]_{≡} are pseudo-closed,

(5) [a]_{E} has finite diameter with respect to ⋁_{n} Φ_{n},

(6) [a]_{E} is pseudo-closed.

Proof. We may assume without loss of generality that X = [a]_{≡}. Then E is orbital.

All the conditions imply that [a]_{E} is pseudo-closed (the first one does by Theorem 3.2, and the others are clearly stronger than (6)).

On the other hand, this condition implies that [a]_{E} has finite diameter (by Theorem 2.12), so all classes have the same, finite diameter (by Fact 2.10), so of course they are pseudo-closed and E is type-definable, and therefore E↾_{Y} is smooth (by Fact 2.7). □

Remark. Corollary 3.3 is, in a way, a strongest possible result. This is to say, there are examples of bounded, F_{σ} and orbital equivalence relations whose Borel cardinality is exactly that of E_{0} (cf. [KPS13, Example 3.3]), so we cannot replace the condition that E↾_{Y} is smooth with some weaker upper bound on Borel cardinality.

For relations refining ≡_{KP}, we may be even more specific.

Corollary 3.4. Assume the language is countable. Suppose E is bounded, F_{σ}, countably supported and orbital on types. Suppose in addition that it refines ≡_{KP}. Then for any a in the domain of E, we have that E↾_{[a]_{≡_{KP}}} is trivial (i.e. total on [a]_{≡_{KP}}) if and only if it is smooth. (In particular, if E is smooth, then it is equal to the restriction of ≡_{KP}.)

Proof. The implication from left to right is trivial. To prove the converse, choose any a in domain of E. The set [a]_{≡_{KP}} is E-saturated (because E refines ≡_{KP}), typedefinable over a and contained in [a]_{≡}, so we can assume without loss of generality that E is defined on [a]_{≡}. Then we can apply Corollary 3.3, which tells us that if E↾_{[a]_{≡_{KP}}} is smooth, then E is type-definable. But in this case E is refined by ≡_{KP} (by [Cas+01, Fact 1.4]), and therefore equal to ≡_{KP} restricted to [a]_{≡}, and so E↾_{[a]_{≡_{KP}}} is trivial. □

We infer an analogous result for invariant subgroups of bounded index of definable groups, whose uncountable version (see Corollary 3.11) will be employed in the final section in the context of definable group extensions.

Corollary 3.5. Assume the language is countable. Suppose that G is a definable group (and therefore countably, and even finitely supported) and H ⊴ G is an invariant, normal subgroup of bounded index, which is F_{σ} (equivalently, generated by a countable family of type-definable sets). Suppose in addition that K ≥ H is a pseudo-closed subgroup of G. Then E_{H}↾_{K} is smooth if and only if H is typedefinable.

Proof. If H is type-definable, then by Lemma 2.19, E_{H} is a type-definable equivalence relation (on a type-definable set), and as such it is immediately smooth by Fact 2.7, and so is its restriction to K.

The proof in the other direction will proceed by contraposition: assume that H is not type-definable. Recall Proposition 2.21: consider, once again, the sorted structure (ℭ, 𝔛, ·).

By Proposition 2.22, H corresponds to a bounded F_{σ} equivalence relation E_{H,X} on 𝔛 (which is not type-definable, since H is not), which is only defined on a single type, and – owing to the assumption that H is normal and Proposition 2.24 – orbital. Evidently K · x_{0} is E_{H,X}-saturated and pseudo-closed, so we can apply Corollary 3.3 to E = E_{H,X} and Y = K ·x_{0}, deducing that E_{H,X}↾_{K·x_{0}} is not smooth, and therefore (by Proposition 2.22) neither is E_{H}↾_{K}. □

3.2. The uncountable case. We intend to formulate the uncountable analogues of Theorem 3.1 and Theorem 3.2, but first we need to introduce some terminology.

Definition. Suppose L′ ⊆ L is some sublanguage, x′ is a tuple of variables and A is a set. Then by L′_{x′}(A) we denote the Lindenbaum-Tarski algebra of (equivalence classes of) L′-formulas with free variables among x′ and parameters from A.

Definition. Suppose we have an F_{σ} equivalence relation E with a normal form ⋁_{n} Φ_{n}(x, y). Suppose in addition that L′ ⊆ L is some sublanguage, x′y′ ⊆ xy is some smaller tuple of variables. Then we define the restriction of the normal form, Φ_{n}↾_{L′_{x′y′} (∅)} as the set of L′_{x′y′}(∅)-consequences of Φ_{n}(x, y), i.e.

Φ_{n}↾_{L′_{x′y′} (∅)} = {φ(x′, y′) ∈ L′_{x′y′}(∅) | Φ_{n}(x, y) ⊢ φ(x′, y′)},

and we define E↾_{L′_{x′y′} (∅)} as the F_{σ} relation given by

a E↾_{L′_{x′y′} (∅)} b ⟺ ℭ ⊧ ⋁_{n} Φ_{n}↾_{L′_{x′y′} (∅)}(a, b).

Remark. For arbitrary L′, x′y′ and E, ⋁_{n} Φ_{n}↾_{L′_{x′y′} (∅)} might not be a normal form (it need not satisfy the triangle inequality, but see the next proposition), but if it is, E↾_{L′_{x′y′} (∅)} is an equivalence relation coarser than E (and with a larger domain) and the metric d′ associated with the restricted normal form satisfies d′ ≤ d.

Proposition 3.6. Given L′, x′y′, we may always extend L′ and x′y′ (without increasing their cardinality by more than |L′| + |x′y′| + ℵ_{0}) to L′′, x′′y′′ in such a way that ⋁_{n} Φ_{n}↾_{L′′_{x′′y′′} (∅)} is a normal form (and consequently, E↾_{L′′_{x′′y′′} (∅)} is an equivalence relation).

Proof. First, we may assume that x′, y′ are symmetric (so that each Φ_{n}↾_{L′_{x′y′} (∅)} is symmetric). Now, for any n < m and any formula φ(x, y) ∈ Φ_{m}(x, y)↾_{L′_{x′y′} (∅)} there are some formulas φ_{1}(x, y), φ_{2}(x, y) in Φ_{n}(x, y) and Φ_{m−n}(x, y), respectively, which witness triangle inequality, that is ⊧ φ_{1} ∧ φ_{2} → φ, and we can add to L′ all the symbols and to x′, y′ all the variables from φ_{1}, φ_{2} (preserving symmetry).

For each pair n < m and formula φ ∈ Φ_{m}(x, y)↾_{L′_{x′y′} (∅)} we add these finitely many symbols and variables (adding no more than |L′| + |x′y′| + ℵ_{0} of them at this step), and we repeat this procedure recursively countably many times (adding no more than ℵ_{0} · (|L′| + |x′y′| + ℵ_{0}) = |L′| + |x′y′| + ℵ_{0} in total). In the end, we have witnesses for all formulas. □

The following result is a theorem from [KMS13], with slightly extended conclusion (which is a part of the proof there).

Theorem 3.7 ([KMS13, Theorem 2.5]). Suppose that X is a regular topological space, ⟨R_{n} | n ∈ N⟩ is a sequence of F_{σ} subsets of X^{2}, Σ is a group of homeomorphisms of X, and O ⊆ X is an orbit of Σ with the property that for all n ∈ N and open sets U ⊆ X intersecting O, there are distinct x, y ∈ O ∩ U with O ∩ (R_{n})_{x} ∩ (R_{n})_{y} = ∅. If X is strong Choquet over O, then there is a function ϕ̃: 2^{<ω} → P(X) such that for any η ∈ 2^{ω} and any n ∈ ω:

• ϕ̃(η↾n) is a nonempty open set,

• \overline{ϕ̃(η↾(n + 1))} ⊆ ϕ̃(η↾n)

Moreover, ϕ(η) = ⋂_{n} ϕ̃(η↾n) = ⋂_{n} \overline{ϕ̃(η↾n)} is a nonempty closed G_{δ} set such that for any η, η′ ∈ 2^{ω} and n ∈ ω:

• if η E_{0} η′, then there is some σ ∈ Σ such that σ · ϕ(η) = ϕ(η′),

• if η(n) ≠ η′(n), then ϕ(η) × ϕ(η′) ∩ R_{n} = ∅, and if η, η′ are not E_{0}-related,

then ϕ(η) × ϕ(η′) ∩ ⋃ R_{n} = ∅.

Proof. As in [KMS13]: what we call ϕ̃(σ) here is γ_{σ} · X_{|σ|} in the proof there. □

We introduce the notion of τ topology which will be crucial in the application in the last section of the paper.

Definition. Suppose X is a topological space. Then by τ we denote the topology on P(X) (i.e. on the family of all subsets of X), or on any subfamily of P(X), generated by subbasis of open sets of the form {A ⊆ X | A ∩ K = ∅} for K ⊆ X closed.

τ topology is weaker than Vietoris topology (it differs in that the sets of the form {A ⊆ X | A ∩ U ≠ ∅} for open U are not included in the subbasis), and it is not, in general, Hausdorff, even when restricted to compact sets. However, we can find some spaces on which it is actually Hausdorff, e.g. ones as in the next simple fact.

Proposition 3.8. Suppose X is a normal topological space (e.g. a compact Hausdorff space) and A is any family of pairwise disjoint, nonempty closed subsets of X. Then A is Hausdorff with τ topology.

Corollary 3.9 (Based on [KMS13, Corollary 2.6]). Let T be any first order theory with language L, M a small model, E a bounded, F_{σ} equivalence relation on a type-definable subset X of a product of sorts compatible with a tuple of variables x. Suppose E has a normal form ⋁_{n} Φ_{n}(x, y). Let Y be an E-saturated subset of X. Finally, suppose we have:

(1) some p ∈ Y_{M},

(2) a countable L′ ⊆ L, a countable M′ ⪯ M↾_{L′} and a countable tuple of variables x′ ⊆ x,

(3) a group Σ of automorphisms preserving M, M′ and all E-classes in Y

setwise. Such that:

(1) the restriction ⋁_{n} Φ_{n}↾_{L′_{x′y′} (∅)}(x, y) is a normal form (i.e. satisfies the triangle inequality),

(2) the topology induced on Y_{M} by L′-formulas with free variables x′ and parameters from M′ is strong Choquet over Σ · p

(3) For every open set U ∋ X in the induced topology and for all N ∈ N, there are some σ ∈ Σ such that σ(p) ∈ U and, letting p′ = p↾_{L′_{x′} (M′)}, we have N < d′_{M′}(σ(p′), p′).

Then there are maps ϕ̃, ϕ into P(Y_{M}) as in Theorem 3.7 with R_{n} = {(p, q) ∈ (Y_{M})^{2} | d_{M}(p, q) ≤ n}.

(Since R_{n} contain the diagonal, it follows that ϕ maps distinct points to disjoint sets, and if η_{1}, η_{2} are E_{0}-inequivalent, then ϕ(η_{1}) × ϕ(η_{2}) ∩ E^{M} = ∅.)

Furthermore, if Y is pseudo-closed, then ϕ is a homeomorphism onto a compact subspace of P(Y_{M}) with τ topology.

Proof. The first part is the same as in [KMS13]; note that thanks to the added condition about the restriction of ⋁_{n} Φ_{n} remaining a normal form, the restricted normal form gives us a metric d′ such that d′ ≤ d (where d is the metric obtained from the original normal form).

The “furthermore” part follows from the fact that ϕ maps distinct points onto disjoint, closed, nonempty subsets of Y_{M}, so in particular, the range rng(ϕ) is a family of disjoint, closed, nonempty subsets of Y_{M}. Since Y_{M} is compact (by Corollary 2.4 and Corollary 2.2), rng(ϕ) is Hausdorff with τ-topology (by Proposition 3.8). We also see that ϕ is injective, so it is a bijection onto rng(ϕ), and since 2^{N} is compact, it is enough to show that ϕ is continuous.

To see that, consider a subbasic open set U = {F | F ∩ K = ∅}, and notice that by compactness, ϕ(η) ∈ U if and only if for some n we have \overline{ϕ̃(η↾n)} ∩ K = ∅, which is clearly an open condition about η. □

Theorem 3.10. Let T be a complete first-order theory, E an orbital on types, bounded, F_{σ} equivalence relation on a λ-supported type-definable set X.

Take some E-saturated and pseudo-closed Y ⊆ X, and assume that there is an element a ∈ Y whose E-class has infinite diameter with respect to some normal form of E, and choose a group Γ witnessing that E↾_{[a]_{≡}} is orbital.

Then there is a model M of size |T | + λ and a function ϕ: 2^{N} → P(Y_{M} ∩ [a]_{≡}) as in the conclusion of Corollary 3.9, i.e. ϕ is a homeomorphic embedding (into P(Y_{M} ∩ [a]_{≡}) with τ-topology) such that for any η, η′ ∈ 2^{N}:

(1) ϕ(η) is a nonempty closed G_{δ},

(2) if η E_{0} η′, then there exists some γ ∈ Γ fixing M setwise such that γ·ϕ(η) = ϕ(η′) (so in particular, their saturations [ϕ(η)]_{E^{M}} , [ϕ(η′)]_{E^{M}} are equal),

(3) if η ≠ η′, then ϕ(η), ϕ(η′) are disjoint,

(4) if η, η′ are not E_{0}-related, then ϕ(η) × ϕ(η′) ∩ E^{M} = ∅.

Proof. We modify the proof of Theorem 5.1 of [KMS13] in a similar way to how we modified Theorem 4.12 there to prove Theorem 3.2.

First, we may assume without loss of generality that Y ⊆ [a]_{≡} (by replacing it with the intersection) and that X = [a]_{≡}; then E is orbital as witnessed by Γ.

Then, we may take a normal form ⋁_{n} Φ_{n}(x, y) as in Proposition 2.11, so that the induced metric satisfies d ≤ d_{L} and [a]_{E} is not d-bounded.

Finally, we want to satisfy the assumptions of Corollary 3.9, which is done in a manner analogous to Theorem 5.1 of [KMS13]: the only difference is that we need to make sure that the restriction of ⋁_{n} Φ_{n}(x, y) is still a normal form, but for that we just need to add another step to the construction to make sure we have all the witnesses (i.e. symbols of L and variables to express necessary formulas) for triangle inequality (like we did in the proof of Proposition 3.6). □

Corollary 3.11. Suppose that G is a definable group and H ⊴ G is an invariant, normal subgroup of bounded index, which is F_{σ} (equivalently, generated by a countable family of type-definable sets). Suppose in addition that K ≥ H is a pseudoclosed subgroup of G. Then, if H is not type-definable, then there is a small model M and homeomorphic embedding ϕ: 2^{N} → P(K_{M}) (where P(K_{M}) is equipped with τ topology) such that for any η, η′ ∈ 2^{N}:

(1) ϕ(η) is a nonempty closed G_{δ},

(2) if η E_{0} η′, then the saturations [ϕ(η)]_{E_{H}^{M}} and [ϕ(η′)]_{E_{H}^{M}} are equal,

(3) if η ≠ η′, then ϕ(η), ϕ(η′) are disjoint,

(4) if η, η′ are not E_{0}-related, then the saturations [ϕ(η)]_{E_{H}^{M}} and [ϕ(η′)]_{E_{H}^{M}} are disjoint.

Proof. Analogous to Corollary 3.5, only instead of Theorem 3.2 we use Theorem 3.10: we get ϕ for E_{H,X} and we compose it with the homeomorphism from Proposition 2.22. □

4. Characterisation of smooth equivalence relations and Borel cardinalities

In this section, we will attempt to characterise the bounded, orbital on types and F_{σ} equivalence relations which are smooth, and in particular, compare smoothness and type-definability. Until the next section, we will assume that the language is countable, along with all the small models and supports of considered equivalence relations (so that the relevant type spaces are Polish).

Firstly, we analyse several examples showing us some of the limitations of this attempt.

4.1. Counterexamples.

Proposition 4.1. Suppose E is a type-definable equivalence relation on a typedefinable set X, and that there are countably many complete ∅-types on X, and infinitely many of them are not covered by singleton E-classes. Then E has a normal form such that the classes of E have unbounded diameter (that is, there is no uniform bound on the diameter).

Proof. Let p_{n} with n > 0 be an enumeration of complete ∅-types on X. Then put (for n > 0)

Φ_{n}(x, y) = (x = y) ∨ (E ∧ ⋁_{m_{1},m_{2}≤n} p_{m_{1}}(x) ∧ p_{m_{2}}(y)) .

It is easy to see that for each n, Φ_{n}(x, y) is a type-definable equivalence relation and Φ_{n} is increasing, so ⋁_{n} Φ_{n}(x, y) is trivially a normal form. In addition, any non-singleton E-class intersecting p_{n} has diameter at least n + 1.

There are infinitely many p_{n} which intersect an E-class which is not a singleton, so in particular, the non-singleton classes have no (finite) uniform bound on diameter. □

Example 4.2. Let T = ACF_{0} be the theory of algebraically closed fields of characteristic 0. Consider E = ≡_{KP} as a relation on ℭ^{2}. The space S_{2}(Q^{alg}) is countable, because T is ω-stable and Q^{alg} is a countable model. This also implies that ≡_{KP} has only countably many classes (on the set of pairs). It is also, of course, smooth, orbital and even type-definable.

Despite being rather well-behaved, E still has a normal form with respect to which the classes have arbitrarily large diameter, which can be seen as follows.

(The set of realisations of) each type of the form tp(q, t/∅) with q ∈ Q and t transcendental is a single, infinite ≡_{KP} -class (because it is the set of realisations of a single type over Q^{alg}), and in particular, it is not covered by singleton classes. Furthermore, S_{2}(∅) is countable (because T is ω-stable). Therefore, by Proposition 4.1, E has a normal form with respect to which its classes have arbitrarily large diameter.

Proposition 4.3. Suppose there is a non-isolated complete ∅-type p_{0} such that p_{0}(ℭ) is not contained in a single class of some definable, bounded (equivalently, with finitely many classes) equivalence relation E. Then the relation

E′(x, y) = (E(x, y) ∨ ¬p_{0}(x)) ∧ (x ≡ y)

is F_{σ} and smooth, but not type-definable.

Furthermore, if E ∩ ≡ is orbital on types, then so is E′.

Proof. A definable and bounded equivalence relation has only finitely many classes, so E′ differs from ≡ only in that one class of ≡ (namely p_{0}(ℭ)) is divided into finitely many pieces. Fix a countable model M and a Borel reduction f: S(M) → X of ≡^{M} as an equivalence relation on S(M) to Δ(X), equality on a Polish space X (which exists because ≡ is smooth, being type-definable).

Let [p_{0}]_{≡}/E = {A_{1}, …, A_{n}}. Then define f̃: S(M) → X ⊔ {1, …, n} (where ⊔ is the disjoint union and {1, …, n} has discrete topology) by

f̃(tp(x/M)) = \begin{cases}f(tp(x/M)) & if x |̸= p_{0}, \\ j & if x ∈ A_{j}. \end{cases}

Then clearly f̃ is Borel and witnesses that E′ is smooth.

E′ is easily seen to be F_{σ}, as it is the intersection of the open (and therefore F_{σ}, as the language is countable) set (E(x, y) ∨ ¬p_{0}(x)) and the closed set (x ≡ y).

It remains to show that E′ is not type-definable. For that, we need the following

Claim. For any formula (without parameters) ψ ∈ p_{0}, there is some x ⊧ p_{0} and x′ |̸= p_{0} such that x′ ⊧ ψ and E(x, x′). In fact, we can find such x′ for any x ⊧ p_{0}.

Proof. The proof is by contraposition: we assume that there are no such x, x′ for ψ, and we will show that p_{0} is isolated. Let

E′′(x, y) = (E(x, y) ∧ ψ(x) ∧ ψ(y)) ∨ (¬ψ(x) ∧ ¬ψ(y)).

Then E′′ is a definable equivalence relation which has finitely many classes (at most 1 more than E) and (by the assumption), p_{0}(ℭ) is a union of E′′-classes, of which there are only finitely many, so p_{0}(ℭ) is definable with some parameters. But since it is invariant, it implies that it is definable without parameters, and therefore p_{0} is isolated.

Once we have some x′ for an x ⊧ p_{0}, we may obtain one for each of them simply by applying automorphisms. □(claim)

Now we choose a sequence φ_{n} of formulas such that ⋀_{n} φ_{n} ⊢ p_{0} and φ_{n+1} ⊢ φ_{n}. Let x_{0}, y_{0} ⊧ p_{0} be such that ¬E(x_{0}, y_{0}) (which we can find because p_{0} is not contained in a single E-class), and let x_{n} be a sequence of elements satisfying φ_{n} but not p_{0}, and simultaneously satisfying E(x_{n}, x_{0}) (this sequence exists by the claim), and let y_{n} be a sequence such that each (x_{0}, x_{n}) is conjugate to (y_{0}, y_{n}) (so that x_{n} ≡ y_{n} and y_{n} ⊧ φ_{n} and E(y_{0}, y_{n})).

Then any limit point of the sequence tp(x_{n}, y_{n}/∅) in S_{2}(∅) is not in E′, even though each tp(x_{n}, y_{n}/∅) is in E′, so E′ is not type-definable.

The “furthermore” part is obvious, since E′ agrees with E ∩ ≡ on p_{0} and is total when restricted to any other type. □

Example 4.4. Consider T = Th(Z, +) (the theory of additive group of integers) and the type p_{0} = tp(1/∅) (the type of an element not divisible by any natural number).

The type p_{0} is not isolated, and it is not contained in a single class of the definable relation E of equivalence modulo 3, while E ∩ ≡ has at most two classes in each complete type, so it is orbital on types due to Corollary 2.16.

In particular – by the preceding proposition – the relation E′(x, y) which says that x ≡ y and they either have the same residue modulo 3 or else they are both divisible by some natural number (i.e. they are not of the same type as 1), is F_{σ}, orbital on types and smooth, but not type-definable.

Example 4.5. Suppose there is some a ∈ ℭ such that ≡_{KP} has two classes on [a]_{≡} (like a = \sqrt{2} for ℭ ⊧ ACF_{0}), so that ≡_{KP} ↾_{[a]_{≡}} ∼_{B} Δ(2). Consider the infinite disjoint union of copies of ℭ, i.e. the multi-sorted structure (ℭ_{n})_{n∈N} where each ℭ_{n} is a distinct sort isomorphic to ℭ (without any relations between elements of ℭ_{n} and ℭ_{m} for n ≠ m). Then consider \overline{a} = (a_{n})_{n∈N} where a_{n} is the element of ℭ_{n} corresponding to a. Then [\overline{a}]_{≡} = ∏_{n}[a_{n}]_{≡} and similarly

(b_{n})_{n} ≡_{KP} (c_{n})_{n} ⟺ ⋀_{n} b_{n} ≡_{KP} c_{n}

(by [Cas+01, Lemma 3.7(iii)]). Now consider the following relation E on [\overline{a}]_{≡}:

(b_{n})_{n} E (c_{n})_{n} ⟺ {n | b_{n} ≢_{KP} c_{n}} is finite.

Then E is refined by ≡_{KP}, but on the other hand,

E ∼_{B} (≡_{KP} ↾_{[a]_{≡}})^{N}/Fin ∼_{B} Δ(2)^{N}/Fin = E_{0}.

(This can be seen e.g. by considering the equivalence relation on [\overline{a}]_{≡}/≡_{KP} induced by E, which is easily seen to be bireducible with E^{M} for a countable model M, using Lemma 2.1 and Proposition 2.3.)

In particular, E is not smooth, it is easy to see that E is F_{σ} (because ≡_{KP} is type-definable and there are countably many finite subsets of N), and it is also orbital, as its classes are just the orbits of the group

{(σ_{n})_{n} ∈ ∏_{n∈N} Aut(ℭ_{n}) | for all but finitely many n, σ_{n} ∈ Aut f_{KP} (ℭ_{n})} .

Additionally, E is only defined on a single type and is not type-definable, so by Corollary 3.3, all its classes have infinite diameter.

Example 4.6. Consider a saturated model K of the theory T = Th(R, +, ·, 0, 1, <) of real closed fields. For each n ∈ N^{+} we have a type-definable equivalence relation Φ_{n}(x, y) = ⋀_{k≥n}(x < k ↔ y < k). Consider the relation E = ⋁_{n} Φ_{n} (with Φ_{0}(x, y) = {x = y}, as before):

• E is an F_{σ} equivalence relation (and since Φ_{n} is an increasing sequence of equivalence relations, it is easy to see that ⋁_{n} Φ_{n} is its normal form).

• E has two classes: the class C_{fin} of elements bounded from above by some natural number, and its complement C_{∞}. Therefore, it is bounded and smooth.

• C_{fin} is a class which is not pseudo-closed (otherwise, by compactness, it

would intersect ⋀_{n} x > n = C_{∞}). This combination of features is possible because E does not refine ≡ (and therefore it is not orbital on types), so we cannot apply Theorem 3.2 to it.

Example 4.7 ([KM13, Example 3.39]). Let T be the theory of an infinite dimensional vector space over F_{2} in the language (+, 0, U_{n})_{n∈N} (i.e. an infinite abelian group of exponent 2), where U_{n} are predicates for independent subspaces of codimension 1 (i.e. subgroups of index 2).

Consider G = ℭ ⊧ T as a definable (additive) group, and let H ≤ G be the intersection of all U_{n}. Then [G : H] = 𝔠, and cosets of H are exactly the types X_{η} = ⋂_{n} U_{n}^{η_{n}}, where η: N → {0, 1}, while U_{n}^{0} = U_{n} and U_{n}^{1} = ℭ \ U_{n}.

Consider the subspaces W_{θ} ≤ G defined as W_{θ} = π^{−1}[ker(θ)], where π: G → G/H is the quotient map, and θ is a nonzero functional G/H → F_{2}. Each θ is uniquely determined by W_{θ} (since its value is 0 on π[W_{θ}] and 1 elsewhere and ker π is contained in all W_{θ}), so there are |(G/H)^{∗}| = 𝔠 > ℵ_{0} distinct W_{θ}, in particular some W = W_{θ} is not definable.

On the other hand, W is invariant, as it is the union of some X_{η} which are type-definable, and [G : W] = 2 (because W has codimension 1), so W is not typedefinable (if it was, its complement would also be type-definable, as it is invariant and a coset of W).

Let us expand ℭ to (ℭ, 𝔛, ·), where 𝔛 is a principal homogeneous space for G. By Proposition 2.21, W induces an invariant equivalence relation E_{W,𝔛} on 𝔛 which has two classes, is orbital by Proposition 2.24 (or Corollary 2.16) and not type-definable by Proposition 2.22.

Remark. The equivalence relation in the previous example is not type-definable, and it is unlikely to even be Borel, as the subspace W is the kernel of an almost arbitrary linear functional, which can be very “wild”. It does show, however, that we need some “definability” hypotheses beyond invariance for the likes of Theorem 2.12.

4.2. Main characterisation theorem.

Theorem 4.8. Let E be an F_{σ}, bounded, orbital on types equivalence relation on a type-definable set X, and d be the invariant metric induced by a normal form of E. Then consider the four conditions:

(1) E classes have uniformly bounded diameter with respect to d.

(2) E is type-definable.

(3) E is smooth.

(4) E classes have finite diameter (equivalently, by Corollary 2.13, they are

pseudo-closed). These conditions are related as follows:

• (1) implies (2),(3),(4),

• (2) implies (3) and (4), but not (1)

• (3) implies (4), but not (2) or (1).

• (4) does not imply (2) or (1). If we assume, in addition, that E refines ≡_{KP} (on X), then conditions (2),(3),(4)

are equivalent (and equivalent to simply E = ≡_{KP} ↾_{X}) and are implied by, but do not imply (1).

If we assume instead that E is only defined on a single complete ∅-type, then all conditions are equivalent.

Proof. For the first part:

• (1) trivially implies (2)

• That (2) implies (3) follows from Fact 2.7.

• That (3) implies (4) follows from (the contraposition of) Theorem 3.2 with Y = X.

• That (2) does not imply (1) follows from Example 4.2.

• That (3) does not imply (2) is demonstrated by Example 4.4.

• Other listed implications (or lack thereof) are logical consequences of the

ones above. To show that the last three conditions are equivalent if E refines KP -type, it

is enough to show that (4) implies (2). But it follows easily from Corollary 2.13. That it does not imply (1) can be seen in Example 4.2.

To show that all four are equivalent if E is defined on a single type, it is enough to notice that (4) implies (1). But this is immediate from the fact that all classes have the same diameter (by Fact 2.10). □

Remarks.

• We have in particular, for bounded E which are F_{σ}, orbital on types and either defined on a single complete type, or refining ≡_{KP}, that E is typedefinable if and only if it is smooth.

• The property that E has only countably many classes implies (2),(3),(4), but not (1) (and is not implied by any of the conditions). (2) follows immediately from Theorem 2.12, while the others follow as a consequence of Theorem 4.8. That having countably many classes does not imply (1), we have seen in Example 4.2. That none of the conditions imply that there is only a countable number of classes can be seen by examining ≡ in a non-small theory.

• Example 4.5 along with Corollary 2.13 show that the condition that E is refined by ≡_{KP} is strictly weaker than all the conditions in Theorem 4.8, even with the added assumption that E is only defined on a single type. (Of course, it is not strictly weaker if we assume that E refines ≡_{KP}, although trivially so.)

4.3. Possible extensions of the characterisation theorem. The picture of logical relations between the four conditions in Theorem 4.8 is almost complete, except for a single implication, raising the following question.

Question 1. In Theorem 4.8, is (3) equivalent to (4)?

A counterexample, if it exists, would be an equivalence relation that is smooth when restricted to any single complete type, but is not on the entirety of its domain.

If, in the first part of Theorem 4.8, we drop the assumption that E is orbital on types (so we allow E to not refine type), then (2) does not imply (4) – as witnessed by Example 2.14 (though (1) certainly still implies the other conditions and (2) implies (3) and that the classes are pseudo-closed).

We can, however, replace the assumption that E is orbital on types by the assumption that it refines type. (Note that by Example 2.17 and Example 2.18, this is not a trivial replacement.) In this case, we do not know whether (3) implies (4) (even if we assume that E refines ≡_{KP} or is defined on a single type), whereas the other implications hold as before.

If we drop the requirement that E is F_{σ}, points (1) and (4) do not make sense. However, the other two do, so another question that arises naturally is the following.

Question 2. Suppose E is a Borel equivalence relation on a type-definable set. Under what conditions does smoothness of E imply that it is type-definable?

Note that we need at least some weak “definability” assumptions, as illustrated by Example 4.7. On the other hand, it is conceivable that some weaker assumptions than Borelness would suffice.

5. Applications to definable group extensions

5.1. Introduction to extensions by abelian groups. This section will show an important application of Corollary 3.11 to definable extensions by abelian groups. More specifically, we deal with short exact sequences of groups of the form

(†)  0 → A → G̃ → G → 0,

where A is an abelian group. In this case, there is a full algebraic description of G̃ in terms of an action of G on A by automorphisms (induced by conjugation in G̃) and a (2-)cocycle h: G^{2} → A (to be defined shortly). We define multiplication on A × G by the formula

(††)  (a_{1}, g_{1}) · (a_{2}, g_{2}) = (a_{1} + g_{1} · a_{2} + h(g_{1}, g_{2}), g_{1}g_{2}).

And a cocycle is defined as follows.

Definition. Let G be a group acting on an abelian group A. A function h: G^{2} → A is a 2-cocycle if it satisfies, for all g, g_{1}, g_{2}, g_{3} ∈ G, the following equations:

h(g_{1}, g_{2}) + h(g_{1}g_{2}, g_{3}) = h(g_{1}, g_{2}g_{3}) + g_{1} · h(g_{2}, g_{3}),
h(g, e) = h(e, g) = e.

The equation (††) endows A × G with group structure – with inverse (a, g)^{−1} = (−g^{−1} · a − h(g^{−1}, g), g^{−1}) – which is compatible with the exact sequence (†), and any G̃ in such a short exact sequence has this form. In this language, the properties of the action and of h reflect the properties of the extension, e.g. central extensions correspond to trivial actions of G on A. More information about this subject (in abstract algebraic terms) can be found in e.g. [Rot02] (section 10.3., and in particular the part up to and including Theorem 10.14).

Definition. We work in an arbitrary given structure. A definable extension of a definable group G by a definable abelian group A is a tuple (G, A, ∗, h), where ∗ is a definable action of G on A by automorphisms and h: G^{2} → H is a definable cocycle. We will also call that the group G̃ = A × G with multiplication defined as in (††).

Remark. The group G̃ introduced above is definable.

In [GK13], the authors have shown that such extensions can, under some additional assumptions, give new examples of definable groups with G^{00} ≠ G^{000}, building upon and extending the intuitions from the first known example with this property, found in [CP12], namely the universal cover of SL_{2}(R). They also pose some questions and conjectures, one of which will be proved at the end of this section. To state their main result, we need the following definition.

Definition. A 2-cocycle h: G^{2} → A is split via f: G → A if for all g_{1}, g_{2} ∈ G we have

h(g_{1}, g_{2}) = df(g_{1}, g_{2}) := f(g_{1}) + g_{1} · f(g_{2}) − f(g_{1}g_{2}),
f(e) = 0.

Now, we recall the main theorem from [GK13] (namely, Theorem 2.2 from there). Later in this section, we will recall and prove two equivalent conjectures from [GK13] which imply that the main technical assumption in this theorem (i.e. assumption (i) below) is not only a sufficient (together with (ii)) but also a necessary condition in a rather general situation.

Theorem 5.1. Let G be a group acting by automorphisms on an abelian group A, where G, A and the action of G on A are ∅-definable in a (saturated) structure G, and let h: G×G → A be a 2-cocycle which is B-definable in G and with finite range rng(h) contained in dcl(B) (the definable closure of B) for some finite parameter set B ⊂ G. By A_{0} we denote the subgroup of A generated by rng(h). Additionally, let A_{1} be a bounded index subgroup of A which is type-definable over B and which is invariant under the action of G. Finally, suppose that:

(i) the induced 2-cocycle \overline{h}_{|G_{B}^{00}×G_{B}^{00}} : G_{B}^{00} × G_{B}^{00} → A_{0}/ (A_{1} ∩ A_{0}) is non-split via B-invariant functions (i.e. there is no B-invariant function f: G_{B}^{00} → A_{0}/(A_{1} ∩ A_{0}) such that \overline{h}↾_{G_{B}^{00}} is split via f),

(ii) A_{0}/ (A_{1} ∩ A_{0}) is torsion free (and so isomorphic with Z^{n} for some natural

n). Then G̃_{B}^{000} ≠ G̃_{B}^{00} (where G̃ is G × A with the group structure defined by (††)).

5.2. Main theorem for definable group extensions. In this subsection, we will fix some (arbitrary) definable extension of a definable group G by an abelian definable group A, the group G̃ = A × G with cocycle h and the corresponding short exact sequence

0 → A → G̃ →^{π} G → 0.

We intend to prove the following theorem, and to that end, we will use Corollary 3.11 and two other results which we will obtain soon.

Theorem 5.2. Suppose H̃ ⊴ G̃ is invariant and of bounded index, generated by a countable family of type-definable sets, contained in a type-definable group \overline{H} ≤ G̃. Put A_{H} := \overline{H} ∩A and H = π[\overline{H}] (both of these are naturally type-definable groups).

Assume that H = π[H̃] and that it acts trivially on A_{H}/H̃ ∩ A, and H̃ ∩ A is type-definable. Then H̃ is type definable.

Until the end of the proof of Theorem 5.2, we are working the in the context (and with the notation) of this theorem.

Lemma 5.3 (generalization of a part of the proof of [GK13, Proposition 2.14]). Given an assignment g ↦ a_{g} such that for g ∈ H we have (a_{g}, g) ∈ H̃, the formula Φ((a, g) · H̃) = a − a_{g} + (H̃ ∩ A) yields a well-defined, injective function \overline{H}/H̃ → A_{H}/(H̃ ∩ A), which does not depend on the choice of a_{g}.

Proof. Consider any (a_{1}, g_{1}), (a_{2}, g_{2}) ∈ \overline{H}.

Since (a_{g_{1}}, g_{1}), (a_{g_{2}}, g_{2}) ∈ H̃, we have

(1) (a_{g_{1}} − g_{1}g_{2}^{−1} · a_{g_{2}} − g_{1} · h(g_{2}^{−1}, g_{2}) + h(g_{1}, g_{2}^{−1}), g_{1}g_{2}^{−1}) =

= (a_{g_{1}}, g_{1})(−g_{2}^{−1}a_{g_{2}} − h(g_{2}^{−1}, g_{2}), g_{2}^{−1}) = (a_{g_{1}}, g_{1})(a_{g_{2}}, g_{2})^{−1} ∈ H̃.

We also have, by simple calculation – for arbitrary g ∈ G and a, a′ ∈ A –

(2)  (a, g)^{−1}(a′, g) = (0, g)^{−1}(−a, e)(a′, e)(0, g) = (g^{−1} · (a′ − a), e).

We want to show that (a_{1}, g_{1})(a_{2}, g_{2})^{−1} ∈ H̃ if and only if (a_{1}−a_{g_{1}})−(a_{2}−a_{g_{2}}) ∈ H̃ ∩ A. The first condition says that

(a_{1} − g_{1}g_{2}^{−1} · a_{2} − g_{1} · h(g_{2}^{−1}, g_{2}) + h(g_{1}, g_{2}^{−1}), g_{1}g_{2}^{−1}) = (a_{1}, g_{1})(a_{2}, g_{2})^{−1} ∈ H̃.

Multiplying it on the left by the inverse of the LHS of (1), and applying (2), we infer that it is equivalent to

(g_{1}g_{2}^{−1})^{−1} ((a_{1} − a_{g_{1}}) − g_{1}g_{2}^{−1} · (a_{2} − a_{g_{2}})) ∈ H̃ ∩ A.

On the other hand, since the action of H on the cosets of H̃ ∩ A is trivial, and g_{1}, g_{2} ∈ H, we can cancel both g_{1}g_{2}^{−1}, and in conclusion the first condition is equivalent to

(a_{1} − a_{g_{1}}) − (a_{2} − a_{g_{2}}) ∈ H̃ ∩ A,

which is just the second condition. Independence of the choice of a_{g} easily follows: for any a_{1}, a_{2} such that (a_{1}, g), (a_{2}, g) ∈ H̃ we have, by the above,

a_{1} − a_{2} = (a_{1} − a_{g}) − (a_{2} − a_{g}) ∈ H̃ ∩ A.  □

Proposition 5.4. Let M be a small model. There exists a continuous function

f: S\overline{H}(M) → A_{H}/(A ∩ H̃) such that for p, q ∈ S\overline{H}(M) we have p EM ; H̃ q ⟺

f(p) = f(q).

Proof. H̃ is F_{σ}, so we can write it as a union ⋃_{i} D_{i} of type-definable subsets of \overline{H}. Without loss of generality, we can assume that D_{i} are all symmetric, contain e and satisfy D_{i}^{2} ⊆ D_{i+1}. Now, consider C_{i} := {g ∈ H | ∃a (a, g) ∈ D_{i}}. Then C_{i} are also type-definable, symmetric, contain e and satisfy C_{i}^{2} ⊆ C_{i+1}, and since π[H̃] = H, there is some n such that H = C_{n} (by Corollary 2.23).

Consider the map f_{1} : \overline{H} → A_{H}/(A ∩ H̃) defined as f_{1}(a, g) = a − a_{g} + (A ∩ H̃), where a_{g} is such that (a_{g}, g) ∈ D_{n} (which exists because C_{n} = H). This is welldefined and does not depend on the choice of a_{g} by Lemma 5.3, which also says that f_{1}(a, g) = f_{1}(a′, g′) if and only if (a, g) EH̃ (a′, g′).

Furthermore, since cosets of A ∩ H̃ are M-invariant (by Corollary 2.5, because A ∩ H̃ is invariant of bounded index), this map factors through S\overline{H}(M), yielding a map f: S\overline{H}(M) → A_{H}/(H̃ ∩ A) which has the desired properties:

• p EM ; H̃ q ⟺ f(p) = f(q), because f_{1}(a, g) = f_{1}(a′, g′) ⟺ (a, g) EH̃

(a′, g′),

• f is continuous: a basic closed set in A_{H}/(H̃ ∩ A) is the quotient of an M-invariant pseudo-closed set, so it is of the form Φ(x, M) + (H̃ ∩ A) for some type Φ. But f^{−1}[Φ(x, M) + (A ∩ H̃)] is just

{tp(a, g/M) | ∃a′ ((a′, g) ∈ D_{n} ∧ a − a′ ∈ Φ(x, M) + (H̃ ∩ A))},

which is clearly a closed set in S\overline{H}(M) (owing to the fact that A ∩ H̃ is type-definable). □

Having proved the above results, we have all but finished the proof of the main theorem.

Proof of Theorem 5.2. Suppose for a contradiction that H̃ is not type-definable. Then let us choose a small model M as in Corollary 3.11 for H = H̃ and K = \overline{H}, so that we have a homeomorphic embedding ϕ: 2^{N} → P(S\overline{H}(M)) (the latter with τ-topology) such that if η E_{0} η′, then the saturations [ϕ(η)]_{EM ; H̃} , [ϕ(η)]_{EM ; H̃} are equal, and otherwise they are disjoint.

The continuous map f from the previous proposition induces a continuous (in τ-topology) mapping f′′: P(S\overline{H}(M)) → P(A_{H}/(A ∩ H̃)), and it is easy to see that f′′ ∘ ϕ maps E_{0}-related points to the same sets, while it mas unrelated points to disjoint closed subsets (because the value of f depends only on the EM ; H̃-class of the argument and any two points from different EM ; H̃-classes are mapped by f to distinct elements).

Consequently, the range of f′′ ∘ ϕ is a compact Polish space: this is because it is Hausdorff by the compactness of A_{H}/(A ∩ H̃) and Proposition 3.8, and the image by a continuous function of a compact Polish space in a Hausdorff space is Polish (as one can easily show that such an image is compact and second-countable). But then f′′ ∘ ϕ is a continuous reduction of E_{0} to equality on rng(f′′ ∘ ϕ), which is a contradiction. □

5.3. Application of the main theorem in group extensions. In [GK13], the authors state the following two equivalent conjectures, which we will prove using Theorem 5.2.

Conjecture 3 ([GK13, Conjecture 2.11]). Suppose we have a definable extension of a definable group G by a definable abelian group A, corresponding to the short exact sequence

0 → A → G̃ → G → 0

with the 2-cocycle h: G^{2} → A. Assume that h has finite range contained in dcl(∅) and that G^{00} = G^{000}.

Then, the conjecture is that for any invariant H̃ ≤ G̃ of bounded index, and such that H̃ ∩ A is type-definable, we have

G̃^{00} ∩ A ⊆ H̃ ∩ A.

Conjecture 4 ([GK13, Conjecture 2.10]). Assume we have G, A, h as in the first paragraph of the previous conjecture. Additionally, let A_{1} be a type-definable subgroup of A of bounded index and invariant under the action of G.

Then, the conjecture is that

G̃^{00} ∩ A ⊆ A_{1} ⟺ G̃^{000} ∩ A ⊆ A_{1}.

Remark. The above conjectures are important mostly for two reasons:

(1) They imply that Corollary 2.8 in [GK13] holds in general (i.e. also when the language is uncountable), that is: in Theorem 5.1, if G_{B}^{00} = G_{B}^{000} and A_{1}^{∗} ⊆ G̃_{B}^{000} ∩ A, then the assumption (i) (about the non-splitting of the modified 2-cocycle) not only implies (under the assumption (ii)), but is also necessary for G̃_{B}^{00} ≠ G̃_{B}^{000}. This is explained in detail in [GK13].

(2) They imply that that, in a rather general context, the quotient G̃^{00}/G̃^{000} is (algebraically) isomorphic to the quotient of a compact group by a finitely generated dense subgroup (this will be revisited at the end of this section).

Remark. The authors of [GK13] actually allow a finite parameter set B over which the cocycle h is definable, they calculate the connected components over this set, and they assume that H̃ is B-invariant in Conjecture 3 and A_{1} is type-definable over B in Conjecture 4. But we may add constants for elements of B to the language, and it changes none of the properties relevant to the previous conjectures, so we assume without loss of generality that B = ∅.

We will also drop the requirement that h has finite range contained in dcl(∅), as it is not needed for the subsequent discussion, which leaves us in the general context of Theorem 5.2, only with the additional assumption that G^{00} = G^{000}.

So, we are working with a definable extensions G̃ of a definable group G by a definable abelian group A. The assumption that G^{00} = G^{000} allows us to make use of the following observation.

Fact 5.5 ([GK13, Remark 2.1(iii)]). If A_{1} is a type-definable, bounded index subgroup of A, invariant under the action of G (i.e. normal in G̃), then G^{00} acts trivially on A/A_{1}.

Proof. A_{1} is G-invariant, so G acts naturally on A/A_{1}, yielding an abstract homomorphism f: G → S(A/A_{1}) (where S(A/A_{1}) is the abstract permutation group). Notice that

ker(f) = {g ∈ G | (∀a) g · a − a ∈ A_{1}}

is a type-definable subgroup of G, and it has bounded index, because S(A/A_{1}) is small. Therefore, it contains G^{00}. □

This leads us to the next corollary.

Corollary 5.6. Suppose H̃⊴G̃ is an invariant subgroup of bounded index, contained in G̃^{00} and F_{σ} (i.e. generated by a countable family of type-definable sets), while G^{00} = G^{000}.

Then H̃ is type-definable (and therefore equal to G̃^{00}) if and only if H̃ ∩ A is type-definable.

Proof. ⇒ is clear. For ⇐ notice that π[H̃] is contained in G^{00}, and contains G^{000}, so it is, in fact, equal to G^{00}, and, by the previous fact, it acts trivially on A/(H̃ ∩ A), so the result follows immediately from Theorem 5.2 with \overline{H} := G̃^{00}. □

And finally, we can prove Conjecture 3.

Corollary 5.7. Suppose H̃ ≤ G̃ is an invariant subgroup of bounded index, and that G^{00} = G^{000}. Suppose in addition that H̃ ∩ A ∩ G̃^{00} is type-definable. Then H̃ ∩ A ⊇ G̃^{00} ∩ A. (In particular, Conjecture 3 holds, without any assumptions on h beyond definability.)

Proof. We intend to apply the preceding corollary, so we need to modify H̃ to satisfy its assumptions.

(1) We can assume without loss of generality that H̃ is normal, because we can replace it with Core(H̃) := ⋂_{g∈G̃} gH̃g^{−1}: the latter is obviously normal and invariant, it has bounded index, because it contains G̃^{000}, and finally, since A, G̃^{00} are normal, we have the identity

Core(H̃) ∩ A ∩ G̃^{00} = Core(H̃ ∩ A ∩ G̃^{00}),

so in particular, the left hand side is type-definable, and hence Core(H̃) satisfies the assumptions of Corollary 5.7 and it is contained in H̃.

(2) Secondly, can assume without loss of generality that H̃ ≤ G̃^{00}, by replacing H̃ with H̃ ∩ G̃^{00}.

(3) Thirdly, we can also assume that H̃ is generated by a countable family of type-definable sets by replacing it with (H̃ ∩ A) · G̃^{000}. This is a normal subgroup, because G̃^{000} is normal, and it is generated by H̃ ∩ A and a type-definable set generating G̃^{000} (which exists by e.g. [GN08, Proposition 3.4]). Moreover, G̃^{000} ≤ H̃, so (H̃ ∩ A) · G̃^{000} ≤ H̃.

(4) Then we can apply the previous corollary to deduce that H̃ = G̃^{00}, so trivially H̃ ∩ A ⊇ G̃^{00} ∩ A. □

As was suggested in [GK13, Remark 2.16], Conjecture 3 implies the next corollary (formulated in a slightly more general form here). By the discussion right after [GK13, Proposition 2.14], this implies that whenever we are in the context of Theorem 5.1, and additionally G_{B}^{000} = G_{B}^{00} and A_{1} ⊆ G̃_{B}^{000} ∩ A, then G̃_{B}^{00}/G̃_{B}^{000} is (abstractly) isomorphic to the quotient of a compact abelian group by a dense finitely generated subgroup – which is analogous to the fact from [CP12]) that for G definable in o-minimal expansions of a real closed field, G^{00}/G^{000} is (abstractly) isomorphic to the quotient of a compact, abelian Lie group by a dense, finitely generated subgroup. This furthers the analogy between [GK13] and [CP12].

Corollary 5.8. Suppose that G̃ is a definable extension of a definable group G by an abelian group A, and that G^{00} = G^{000}.

Let A_{1} ≤ G̃^{000} ∩ A be a G-invariant, type-definable subgroup of A of bounded index. Then (G̃^{000} ∩ A) /A_{1} is dense in (G̃^{00} ∩ A) /A_{1} (with the logic topology).

Proof. Let A_{2} be the preimage of the closure of (G̃^{000} ∩ A)/A_{1} by the quotient map π: A → A/A_{1}. Then A_{2} is a type-definable (over ∅) subgroup of A, which contains G̃^{000} ∩ A.

Then, H̃ := A_{2} · G̃^{000} is an invariant subgroup of G̃ of bounded index, and H̃ ∩ A = A_{2} is type-definable, as is H̃ ∩ A ∩ G̃^{00}, so by Corollary 5.7, A_{2} = H̃ ∩ A ⊇ G̃^{00} ∩ A, and therefore A_{2}/A_{1} – the closure of (G̃^{000} ∩ A)/A_{1} in A/A_{1} – contains (G̃^{00} ∩ A)/A_{1}, which was to be shown. □

References

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Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

E-mail address, K. Krupiński: kkrup@math.uni.wroc.pl

E-mail address, T. Rzepecki: tomasz.rzepecki@math.uni.wroc.pl