G-COMPACTNESS AND GROUPS

JAKUB GISMATULLIN AND LUDOMIR NEWELSKI

Abstract. Lascar described E_{KP} as a composition of E_{L} and the topological closure of E_{L} ([1]). We generalize this result to some other pairs of equivalence relations.

Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M′ consisting of M and X as two sorts, where X is an affine copy of G and in M′ we have the structure of M and the action of G on X. We prove that the Lascar group of M′ is a semi-direct product of the Lascar group of M and G/G_{L}. We discuss the relationship between G-compactness of M and M′. This example may yield new examples of non-G-compact theories.

1. Introduction

Let T be a complete theory in language L. We work within a monster model ℭ ⊧ T. A model M ⊧ T is small if M ≺ ℭ and |M| = |T |. If X is a subset of a topological space, then by int(X) we denote its interior and by cl(X) its closure. We recall some well known facts about the Lascar Group and Lascar strong types (see [1, 9]). The group of Lascar strong automorphisms is defined by:

Autf_{L}(ℭ) = ⟨ Aut(ℭ/M): M is a small model ⟩,

and the Lascar (Galois) group of T by:

Gal_{L}(T) = Aut(ℭ)/ Autf_{L}(ℭ).

This definition does not depend on the choice of the monster model ℭ of T (it is enough that ℭ is |T |^{+}-saturated and |T |^{+}-strongly homogeneous). We say that a, b ∈ ℭ^{k} (k < |T |^{+}) have the same Lascar strong type, and write E_{L}(a, b), if there exists f ∈ Autf_{L}(ℭ) such that a = f(b). Thus E_{L} is a ∅-invariant and bounded equivalence relation on every sort ℭ^{k} (because if a ≡_{M} b for some small M ≺ ℭ, then E_{L}(a, b), so

|ℭ^{k}/E_{L}| ≤ |S_{k}(M)| ≤ 2^{|T |}).

Definition 1.1. A symmetric formula φ(x, y) ∈ L_{k+k}(∅) is thick if for some n < ω, for every sequence (a_{i})_{i<n} there exist i < j < n such that φ(a_{i}, a_{j}). By Θ we denote the conjunction of all thick formulas:

Θ(x, y) = ⋀_{φ thick} φ(x, y).

In the above definition we can equivalently take an infinite sequence (a_{i})_{i<ω}. If φ and θ are thick, then ψ(x, y) = φ(y, x) and φ ∧ θ are also thick (this follows from the Ramsey theorem). Θ is a ∅-invariant relation (not necessarily an equivalence relation) and if Θ(a_{0}, a_{1}), then we can extend (a_{0}, a_{1}) to an order indiscernible sequence (a_{i})_{i<ω}.

2000 Mathematics Subject Classification. 03C60, 22C05, 20F28, 03H05.

Key words and phrases. strong types, G-compactness, regular action, automorphisms groups.

On the other hand if (a_{i})_{i<ω} is a 2-indiscernible sequence, then φ(a_{0}, a_{1}) for every thick φ.

Lemma 1.2. [9, Lemma 7] (i) If Θ(a, b), then there is a small M such that a ≡_{M} b.

(ii) If for some small M we have a ≡_{M} b, then Θ^{2}(a, b), i.e. there is c such that

Θ(a, c) ∧ Θ(c, b). (iii) E_{L} is the transitive closure of Θ.

If π is a type over ∅, then we can define thick formulas on π(ℭ) and their conjunction Θ_{π} similarly as in the above definition. Moreover, the last remark also holds for Θ_{π}, so E_{L}|_{π(ℭ)} is the transitive closure of Θ_{π}. One can prove that E_{L}|_{π(ℭ)} is the finest bounded ∅-invariant equivalence relation on π(ℭ).

There is a compact (not necessarily Hausdorff) topology on the group Gal_{L}(T). Let M and N be arbitrary small models and let

S_{M}(N) = {tp(M′/N): tp(M′) = tp(M)}

be a closed subset of S_{|T |}(N). Thus S_{M}(N) carries a compact subspace topology. The quotient map j: Aut(ℭ) → Gal_{L}(T) factors as j = ν ∘ μ, where μ: Aut(ℭ) → S_{M}(N) maps f to tp(f(M)/N), and μ: S_{M}(N) → Gal_{L}(T) maps tp(f(M)/N) to an appropriate coset of Autf_{L}(ℭ), so we have the following commutative diagram:

\xymatrix{Aut(ℭ) \ar@{->>}[rr]^{j=ν∘μ} \ar@{->>}[dr]_{μ} &  & Gal_{L}(T) = Aut(ℭ)/ Autf_{L}(ℭ) \\  & S_{M}(N) \ar@{->>}[ur]_{ν}}

We can induce topology on Gal_{L}(T) from ν, i.e. X ⊆ Gal_{L}(T) is closed if and only if its preimage ν^{−1}[X] is closed in S_{M}(N). It can be easily seen that this definition of topology does not depend on the choice of small models M and N ([9, Theorem 4]). With this topology Gal_{L}(T) becomes a compact topological group. We say that T is G-compact when Gal_{L}(T) is Hausdorff. If we consider Aut(ℭ) with the usual topology of pointwise convergence, then all the maps in the diagram are continuous. However ν need not be open, instead ν satisfies some weak kind of openness.

Theorem 1.3. [9, Lemma 12] For p ∈ S_{M}(N) define its Θ-neighbourhood as:

[p]_{Θ} = {q ∈ S_{M}(N): p(x) ∪ q(y) ∪ Θ(x, y) is consistent }.

If we take an arbitrary point p ∈ S_{M}(N) and subset U ⊆ S_{M}(N) such that [p]_{Θ} ⊆ int(U), then ν(p) ∈ int(ν[U]).

The relation E_{L} is ∅-invariant, so we may consider E_{L} as a subset of S_{|T |+|T |}(∅). Using this, we define the relation \overline{E_{L}} as cl(E_{L}). \overline{E_{L}} is ∅-invariant and contains E_{L}. There exists the finest bounded ⋀-definable over ∅ equivalence relation, denoted by E_{KP} and known as equality of Kim-Pillay strong types (there is also an appropriate group of automorphisms Autf_{KP}(ℭ) such that E_{KP} (a, b) if and only if for some f ∈ Autf_{KP}(ℭ), a = f(b)). The next theorem describes some relationship between E_{KP} , Θ and E_{L}.

Theorem 1.4. [1, Corollary 2.6] E_{KP} = Θ ∘ \overline{E_{L}}

An attempt to understand the proof of this theorem was a starting point of this paper. In particular it was puzzling what properties of E_{L}, E_{KP} and Θ are responsible for the relationship described in Theorem 1.4. It turnes out that the important point here is that both E_{L} and E_{KP} are orbit equivalence relations with respect to some groups of automorphisms of ℭ. We elaborate on this in Section 2. We generalize Theorem 1.4 there and give a new proof of it based on Theorem 1.3. Also in Section 2 we generalize some results about Lascar, Kim-Pillay and Shelah strong types.

Section 3 contains a model-theoretic analysis of a structure N = (M, X, ·), where M is a given stucture and X is affine copy of some group G definable in M. We describe the group of automorphisms of N as a semi-direct product of G and the group of automorphisms of M. In particular we reduce the question of G-compactness of N to the question of ⋀-definability of certain subgroup G_{L} of G. This motivates us to look for examples of G, where G_{L} is not ⋀-definable.

In Section 4 we verify that G_{L} is ⋀-definable in several cases, e.g. when M is small or simple or o-minimal and G is definable compact.

In Section 5 we provide an example where a subgroup of G, similar in some sense to G_{L}, is not ⋀-definable, and also an example of a group G that is not G-compact.

We assume that the reader is familiar with basic notions of model theory.

The results in Sections 2, 3 and 4 are due to the first author, the proof of Lemma 3.7(1) and the examples in Section 5 are due to the second author.

2. Orbit equivalence relations

In this section G is always a subgroup of Aut(ℭ). We can consider the orbit equivalence relation E_{G} defined as follows: E_{G}(a, b) if and only if there is some f ∈ G with a = f(b), where a and b are tuples of elements of ℭ of length ≤ |T |, such tuples are called small. In this paper we consider E_{G} as an equivalence relation on the sets of small tuples of elements of various sorts of ℭ.

The results of this section are concerned with various properties of relations of the form E_{G}. Our motivation is based on the observation that almost all important equivalence relations in model theory (e.g. E_{L}, E_{KP} and E_{Sh}) are of this form.

Some statements from the next proposition are probably well known (see [1, 5, 7, 9]).

Lemma 2.1. (i) Let M be an arbitrary small model, then

G · Aut(ℭ/M) = {f ∈ Aut(ℭ): E_{G}(M, f(M))}.

(ii) The relation E_{G} is ∅-invariant on every sort if and only if for every small M ≺ ℭ and every F ∈ Aut(ℭ)

G ⊆ G^{F} · Aut(ℭ/M).

In particular if G contains ⋃_{F ∈Aut(ℭ)} Aut(ℭ/F [M]) for some small M, then E_{G} is ∅-invariant if and only if G ◁ Aut(ℭ).

(iii) If G has bounded index in Aut(ℭ), then E_{G} is bounded and E_{L} ⊆ E_{G}. If E_{G} is ∅-invariant bounded G ◁ Aut(ℭ) and G contains Aut(ℭ/M) for some small M, then G has bounded index in Aut(ℭ).

(iv) Let j: Aut(ℭ) ⟶ Gal_{L}(T) be the quotient map and assume that Autf_{L}(ℭ) ⊆ G.

(a) j[G] is closed in Gal_{L}(T) if and only if E_{G} is ⋀-definable over any small model. If G ◁ Aut(ℭ), then ⋀-definability is over ∅.

(b) j[G] is open in Gal_{L}(T) if and only if G = Aut(ℭ/e) for some e ∈ acl^{eq}(∅) (i.e. e = \overline{m}/F for some ∅-definable finite equivalence relation F on some ℭ^{n}, n < ω).

Proof. (i) Easy.

(ii) Without loss of generality we may work with small models, because every tuple a may be extended to small model M. Take an arbitrary small M ≺ ℭ, g ∈ G and F ∈ Aut(ℭ). Then E_{G}(M, g(M)). Assume that E_{G} is ∅-invariant. Then E_{G}(F (M), F (g(M))) holds, so for some g′ ∈ G, F (g(M)) = g′(F (M)). Thus F^{−1} ∘ g^{′−1} ∘ F ∘ g ∈ Aut(ℭ/M), so g ∈ g^{′F} ∘ Aut(ℭ/M) ⊆ G^{F} ∘ Aut(ℭ/M). The other implication is similar.

For the second statement of (ii) assume that G ⊆ G^{F} · Aut(ℭ/M). Then conjugating by F^{−1} we obtain

G^{F^{−1}} ⊆ G · Aut(ℭ/F [M]) = G,

for an appropriate small model M.

(iii) If G has bounded index in Aut(ℭ), then there is a normal subgroup H ◁ Aut(ℭ) of bounded index, with H ⊆ G (an intersection of boundedly many conjugates of G). Thus E_{H} is bounded and invariant, so E_{L} ⊆ E_{H} ⊆ E_{G}.

For the second statement we use (i) to conclude that G = Aut(ℭ/⌜M/E_{G}⌝). G has bounded index, because M/E_{G} has boundedly many conjugates.

(iv) Note that j^{−1}[j[G]] = G·Autf_{L}(ℭ) = G, thus μ[G] = ν^{−1}[j[G]] (because j = ν∘μ).

(a) ⇒: Let M be an arbitrary small model. If j[G] is closed in Gal_{L}(T), then μ[G] = ν^{−1}[j[G]] = {tp(M′/M): Φ(M′, M)} for some type Φ(x, y) over ∅. We have that

E_{G}(a, b) ⟺ (∃f ∈ Aut(ℭ)) (a = f(b) ∧ Φ(f(M), M)) ,

and thus E_{G} is ⋀-definable over M:

E_{G}(a, b) ⟺ (∃z)(tp(b, M) = tp(a, z) ∧ Φ(z, M)).

⇐: There is a type Φ(x, y) over M such that

E_{G}(a, b) ⟺ Φ(a, b).

Since μ[G] = ν^{−1}[j[G]] it is enough to prove that μ[G] is closed in S_{M}(M). This is clear, because:

μ[G] = {tp(g(M)/M): g ∈ G} = {tp(M′/M): Φ(M′, M)}.

(b) ⇒: First we deal with the case where G ◁ Aut(ℭ). Since Gal_{L}(T) is a compact topological group, j[G] has finite index in Gal_{L}(T), hence it is closed. By (iva) E_{G} is ∅-⋀-definable. Also G has finite index in Aut(ℭ). It follows that E_{G} has finitely many classes on tp(M)(ℭ) (the set of realisations of type tp(M)) and from (i) we have G = Aut(ℭ/(M/E_{G})). Hence there are a finite ∅-definable equivalence relation F and \overline{m} ⊂ M such that G = Aut(ℭ/(\overline{m}/F)).

Now we deal with the general case, so G < Aut(ℭ) need not be normal. However, still G has finite index in Aut(ℭ). Hence there is a normal subgroup H ◁ Aut(ℭ) contained in G and such that j[H] is open (an intersection of finitely many conjugates of G). We may apply the first case to H. We get an e ∈ acl^{eq}(∅) such that H = Aut(ℭ/e). An element e has finitely many conjugates, so e′ = ⌜{g · e: g ∈ G}⌝ ∈ acl^{eq}(∅). Now it is obvious that G = Aut(ℭ/e′).

⇐: The subset ν^{−1}[j[G]] = μ[G] = \{tp(f(M)/M): F (\overline{m}, f(\overline{m})), f ∈ Aut(ℭ)\} of S_{M}(M) is clopen. □

Problem 2.2. Consider an equivalence relation E on sorts of ℭ which is ∅-invariant. Then we can build the following growing sequence of ∅-invariant relations:

(i) E_{0} = E,

(ii) E_{1} = cl(E) in S_{k+k}(∅),

(iii) for 1 ≤ α ∈ Ord let • E_{α+1} = cl(transitive closure of E_{α}),

• if α ∈ Lim, then E_{α} = ⋃_{λ<α} E_{λ}.

Take E_{∞} = ⋃_{α∈Ord} E_{α}. Then clearly E_{∞} is the finest type definable equivalence relation which extends E, so we may ask the question: what is the first α_{E} for which E_{α_{E}} = E_{∞}? If E = E_{G}, where Autf_{L}(ℭ) ⊆ G ◁ Aut(ℭ), then from the next Theorem 2.3(ii) we conclude that α_{E} ≤ 2.

It can be proved that Autf_{KP}(ℭ) = j^{−1}[cl(id_{Gal_{L}(T)})]. Recall that E_{KP} = E_{Autf_{KP}(ℭ)} is the finest bounded ⋀-definable over ∅ equivalence relation. The next Theorem 2.3(i) generalizes this remark and Theorem 1.4 to an arbitrary group of automorphisms containing Autf_{L}(ℭ).

Theorem 2.3. Let Autf_{L}(ℭ) ⊆ G < Aut(ℭ) and consider \overline{\overline{G}} = j^{−1}[cl(j[G])]. Then

(i) On each sort of ℭ the relation E\overline{\overline{G}} is the finest bounded ⋀-definable over any small model equivalence relation which extends E_{G}.

(ii) If additionally G ◁ Aut(ℭ), then

E\overline{\overline{G}} = Θ ∘ \overline{E_{G}},

where \overline{E_{G}} is cl(E_{G}) in S_{k+k}(∅).

Proof. (i) Let E be a ⋀-definable over M equivalence relation and E_{G} ⊆ E. Take an arbitrary f ∈ \overline{\overline{G}} and a small tuple b. We have to prove that E(f(b), b). Consider the following set

H = {f ∈ Aut(ℭ): E(f(b), b)}

(H is not necessarily a group, because E is not necessarily ∅-invariant). It is enough to show that \overline{\overline{G}} ⊆ H.

Note that j^{−1}[j[H]] = Autf_{L}(ℭ) · H = H, because for f ∈ Autf_{L}(ℭ), h ∈ H we have E(h(b), b) and E(f(h(b)), h(b)) (E_{L} ⊆ E), so E(f(h(b)), b) and f ∘ h ∈ H.

Since E_{G} ⊆ E we have G ⊆ H, so we must only prove that cl(j[G]) ⊆ j[H] (because j^{−1}[j[H]] = H). The proof is completed by showing that j[H] is closed in Gal_{L}(T). This follows from the fact that the set

ν^{−1}[j[H]] = μ[H] = {tp(f(M′)/M′): E(f(b), b), f ∈ Aut(ℭ)}

is closed in S_{M′}(M′), where Mb ⊆ M′ ≺ ℭ.

(ii) The relation E\overline{\overline{G}} is ⋀-definable over ∅, so E\overline{\overline{G}} is a closed subset of S_{k+k}(∅), thus \overline{E_{G}} ⊆ E\overline{\overline{G}}. This gives Θ ∘ \overline{E_{G}} ⊆ E\overline{\overline{G}}.

Now we prove that E\overline{\overline{G}} ⊆ Θ∘\overline{E_{G}}. Assume that a, b are small tuples such that E\overline{\overline{G}}(a, b),

i.e. a = f(b) for some f ∈ \overline{\overline{G}}. Without loss of generality we may assume that b = M, for some small M ≺ ℭ, so a = f(M). Let p = μ(f) = tp(f(M)/M). Then ν(p) = j(f) ∈ cl(j[G]) and

[p]_{Θ} ∩ cl(ν^{−1}[j[G]]) ≠ ∅,

because otherwise [p]_{Θ} ⊆ int(ν^{−1}[j[G]^{c}]), and from Theorem 1.3

ν(p) ∈ int(ν[ν^{−1}[j[G]^{c}]]) = int(j[G]^{c}) = cl(j[G])^{c},

a contradiction.

Let q ∈ [p]_{Θ} ∩ cl(ν^{−1}[j[G]]). There is some c = M′ ⊧ q such that Θ(f(M), M′) and q = tp(M′/M) is in

cl(ν^{−1}[j[G]]) = cl(μ[G]) = cl{tp(g(M)/M): g ∈ G}
= cl{tp(g(M)/M): E_{G}(g(M), M)}.

Finally tp(M′, M) ∈ cl(E_{G}) = \overline{E_{G}}, and we obtain that Θ(a, c) and \overline{E_{G}}(c, b). □

Now we consider the relation E_{Sh} of equality of Shelah strong types:

E_{Sh} = ⋂{E: E is a ∅-definable finite equivalence relation}.

It can be proved that E_{Sh} = E_{j^{−1}[QC]}, where QC ◁ Gal_{L}(T) is the intersection of all open subgroups of Gal_{L}(T) (the quasi-connected component). When Gal_{L}(T) is Hausdorff (i.e. T is G-compact) then QC is just the connected component of Gal_{L}(T).

In the next proposition we generalize this property of E_{Sh}, but first we need a definition: if A ⊆ Gal_{L}(T), then by QC(A) we denote the following set

⋂{H < Gal_{L}(T): A ⊆ H and H is open}.

Proposition 2.4. If H < Gal_{L}(T), then E_{j^{−1}[QC(H)]} is the intersection of all ∅-definable finite equivalence relations which extend E_{j^{−1}[H]}:

E_{j^{−1}[QC(H)]} = ⋂{E: E is a ∅-definable finite e.r. and E_{j^{−1}[H]} ⊆ E}.

Moreover j^{−1}[QC(H)] is equal to the group of all f ∈ Aut(ℭ), satisfying

E_{j^{−1}[QC(H)]}(a, f(a))

for arbitrary small tuple a.

Proof. First we prove the equality of relations. (⊆) Assume that small tuples a, b are E_{j^{−1}[QC(H)]} equivalent, so a = f(b) for some f ∈ j^{−1}[QC(H)], and E is a ∅-definable finite equivalence relation extending E_{j^{−1}[H]}. Define

G′ = {f ∈ Aut(ℭ): E(f(b), b)} = Aut(ℭ/(b/E)).

Then H ⊆ j[G′] and j[G′] is open as a subset of Gal_{L}(T) (Lemma 2.1(iv)(b)). Therefore QC(H) ⊆ j[G′] and

f ∈ j^{−1}[QC(H)] ⊆ j^{−1}[j[G′]] = G′ · Autf_{L}(ℭ) = G′,

so E(a, b) holds.

(⊇) Let QC(H) = ⋂{G_{i}: i ∈ I}. Using Lemma 2.1(iv)(b) we can find (e_{i})_{i∈I} ⊆ acl^{eq}(∅) such that j^{−1}[G_{i}] = Aut(ℭ/e_{i}). Then j^{−1}[QC(H)] = Aut(ℭ/{e_{i}}_{i∈I}). We can assume that e_{i} = m_{i}/F_{i} for some ∅-definable finite equivalence relations F_{i}. Assume that (a, b) belongs to

⋂{E: E is a ∅-definable finite e.r. and E_{j^{−1}[H]} ⊆ E}.

We have to find f ∈ j^{−1}[QC(H)] for which b = f(a). It suffices to prove that the following type in variables (y_{i})_{i∈I} is consistent:

“ tp(b, y_{i})_{i∈I} = tp(a, m_{i})_{i∈I}” ∧ ⋀_{i∈I} F_{i}(m_{i}, y_{i}).

Let φ(x, x_{1}, …, x_{n}) ∈ tp(a, m_{i})_{i∈I}. It is enough to show that

ψ(b, m_{1}, …, m_{n}) = (∃y_{1}, …, y_{n}) (φ(b, y_{1}, …, y_{n}) ∧ ⋀_{1≤i≤n} F_{i}(m_{i}, y_{i}))

holds. The formula ψ = ψ(x, m_{1}, …, m_{n}) is almost over ∅ (because m_{1}, …, m_{n} ∈ acl(∅)). Let ψ_{1}, …, ψ_{k} be all conjugates of ψ over ∅ and take

A(x, y) = ⋀_{1≤i≤k}(ψ_{i}(x) ↔ ψ_{i}(y)).

A is a ∅-definable finite equivalence relation and E_{j^{−1}[H]} ⊆ A (because j^{−1}[H] ⊆ j^{−1}[QC(H)] = Aut(ℭ/(m_{i}/F_{i}))). Therefore A(a, b) and we know that ψ(a, m_{1}, …, m_{n}) holds, so ψ(b, m_{≤n}) also holds.

Now we prove the second part of the proposition. Let G′ be the group of all automorphisms preserving E_{j^{−1}[QC(H)]}. Inclusion j^{−1}[QC(H)] ⊆ G′ is obvious.

(⊇) Let g ∈ G′ and a = M be a small model. Then E_{j^{−1}[QC(H)]}(M, g(M)), so g(M) = f(M) for some f ∈ j^{−1}[QC(H)]. Thus j(g) = j(f) (because gf^{−1} ∈ Autf_{L}(ℭ)) and j(f) ∈ QC(H). Therefore g ∈ j^{−1}[QC(H)]. □

3. An Example

Let M be an arbitrary structure in which we have a ∅-definable (interpretable) group G. In this section we consider the following two sorted structure: N = (M, X, ·), where

• X and M are disjoint sorts,

• ·: G × X → X is a regular (free and transitive) action of G on X i.e. X is an affine copy of G,

• on M we take its original structure. This structure was already considered e.g. in [9, 7]. Our study of N is based on ideas

from [9, Section 7].

In this section we describe various groups of automorphisms of N in terms of appropriate groups of automorphisms of M and groups related to G. We also give a description of the relations E_{L}, E_{KP} and E_{Sh} on the sort X of N. In particular, in Corollary 3.6 we prove that G-compactness of N is equivalent to G-compactness of M and ⋀-definability of certain subgroup G_{L} of G. Thus constructing a group G where the subgroup G_{L} is not ⋀-definable may yield a new example of a non-G-compact theory.

Fix an arbitrary point x_{0} from X and take N^{∗} = (M^{∗}, X^{∗}, ·), a monster model extending N. Then G ⊆ G^{∗} and X = G · x_{0} ⊆ G^{∗} · x_{0} = X^{∗}.

The group G^{∗} acts on itself in two different, but commuting ways, the first one is by left translation (g, h) ↦ gh, and the second one by the following rule (g, h) ↦ hg^{−1}. We define homomorphic embeddings of automorphism groups:

\overline{·}: Aut(M^{∗}) ↪ Aut(N^{∗}), \overline{·}: G^{∗} ↪ Aut(N^{∗}).

Let h ∈ G^{∗}, f ∈ Aut(M^{∗}), g ∈ G^{∗}. We define \overline{f}, \overline{g} ∈ Aut(N^{∗}) by:

\overline{f}|_{M^{∗}} = f, \overline{f}(h · x_{0}) = f(h) · x_{0},
\overline{g}|_{M^{∗}} = id_{M^{∗}}, \overline{g}(h · x_{0}) = (hg^{−1}) · x_{0}.

It is easy to verify the following laws: for f ∈ Aut(M^{∗}), g ∈ G^{∗} we have

\overline{f} ∘ \overline{g} = \overline{f(g)} ∘ \overline{f}, \overline{g} ∘ \overline{f} = \overline{f} ∘ \overline{f^{−1}(g)}.

Using these embeddings we can identify Aut(M^{∗}) and G^{∗} with their images in Aut(N^{∗}) and conclude that G^{∗} ◁ Aut(N^{∗}). In fact we will prove that Aut(N^{∗}) is a semi-direct product of G^{∗} and Aut(M^{∗}).

There are two different actions of the group G^{∗} on the set X^{∗}: the first one comes from the above embedding

\overline{g}(h · x_{0}) = (hg^{−1}) · x_{0}

(it is definable over x_{0}). The second one comes from the regular action

g · (h · x_{0}) = (gh) · x_{0}.

If A ⊆ G^{∗} satisfies hA^{−1} = Ah, then the orbits of h · x_{0} under both actions coincide: \overline{A}(h · x_{0}) = A · (h · x_{0}) (in this case we just write A · (h · x_{0})).

In order to describe properties of N^{∗} in terms of M^{∗} and G^{∗} we need the next definition.

Definition 3.1. For a group G and a binary relation E on G we define the set of Ecommutators X_{E} = {a^{−1}b: a, b ∈ G, E(a, b)} and the E-commutant G_{E} as the subgroup of G generated by X_{E}

G_{E} = ⟨X_{E}⟩ < G.

Remark 3.2. If E = E_{H} for some H < Aut(G, ·), then G_{E_{H}} ◁ G. If E is ∅-invariant, then X_{E} and G_{E} are also ∅-invariant. If E is bounded, then G_{E} has bounded index in G, moreover [G : G_{E}] ≤ |G/E|.

Proof. Let a, x ∈ G and h ∈ H. Then

(X_{E_{H}})^{x} ∋ (a^{−1}h(a))^{x} = (ax)^{−1}h(a)x = ((ax)^{−1}h(ax))(h(x)^{−1}x) ∈ X_{E_{H}}^{2}.

The last statement follows from the observation: if a^{−1}b ∉ G_{E}, then ¬E(a, b). □

The following example justifies the names “E-commutators” and “E-commutant” from the previous definition. Let E be the conjugation relation in G i.e. E = E_{Inn(G)} (where Inn(G) is the group of inner automorphisms of G). Then X_{E} is the set of all commutators and G_{E} = [G, G].

In the case where E = E_{L} [E = E_{KP} , E_{Sh}, respectively] we just write X_{L} and G_{L} [X_{KP}, X_{Sh}] instead of X_{E_{L}} and G_{E_{L}} [X_{E_{KP}}, X_{E_{Sh}}]. Note that G_{L} is generated by X_{Θ}.

In the next proposition we describe Aut(N^{∗}), Autf_{L}(N^{∗}) and Gal_{L}(Th(N)) as semidirect products of automorphisms groups of M^{∗} and appropriate groups associated with G.

Proposition 3.3. (1) Aut(N^{∗}) = G^{∗} ⋊Aut(M^{∗}), more precisely: for F ∈ Aut(N^{∗}),

F = \overline{g} ∘ \overline{f}, where f = F |_{M^{∗}} and F (x_{0}) = g^{−1} · x_{0}.

(2) Let (N′, X′) ≺ (N^{∗}, X^{∗}) and X′ = G′ · (h_{0} · x_{0}) for some h_{0} · x_{0} ∈ X′. Then

F ∈ Aut(N^{∗}/N′) ⟺ (∃f ∈ Aut(M^{∗}/M′)) (F = \overline{f}\overline{h_{0}}) .

(3) Autf_{L}(N^{∗}) = G_{L}^{∗} ⋊ Autf_{L}(M^{∗}) and Gal_{L}(Th(N)) = G^{∗}/G_{L}^{∗} ⋊ Gal_{L}(Th(M)).

Proof. (1) Let F ∈ Aut(N^{∗}) and f = F |_{M^{∗}}. Then F \overline{f}^{−1} is the identity on M^{∗}, and on X^{∗} = G^{∗} · x_{0} we have:

F \overline{f}^{−1}(h · x_{0}) = F (f^{−1}(h) · x_{0}) = h · F (x_{0}) = h · (g^{−1} · x_{0}) = \overline{g}(h · x_{0}),

for some g ∈ G^{∗}. Thus F = \overline{g} ∘ \overline{f}. The group Aut(M^{∗}) acts on G^{∗} by conjugation, so for g ∈ G^{∗} and f ∈ Aut(M^{∗}), \overline{g}\overline{f} = \overline{f(g)} ∈ G^{∗}. It is clear that G^{∗} ∩ Aut(M^{∗}) = \{0\}.

(2) (⇐) It is clear that F |_{M′} = id_{M′}. Using the fact that f|_{M′} = id_{M′} we get for h′ ∈ X′:

\overline{f}\overline{h_{0}}(h′h_{0} · x_{0}) = \overline{h_{0}^{−1}} ∘ \overline{f}(h′ · x_{0}) = \overline{h_{0}^{−1}}(h′ · x_{0}) = h′h_{0} · x_{0}.

Thus \overline{f}\overline{h_{0}}|_{X′} = id_{X′}.

(⇒) Let f = F |_{M^{∗}}. Then f = \overline{f}\overline{h_{0}}|_{M^{∗}} ∈ Aut(M^{∗}/M′). By assumptions

h_{0} · x_{0} = F (h_{0} · x_{0}) = F (h_{0}) · F (x_{0}) = f(h_{0}) · F (x_{0}),

and then F (x_{0}) = f(h_{0}^{−1})h_{0} · x_{0}. By (1), F = \overline{h_{0}^{−1}f(h_{0})} ∘ \overline{f} = \overline{h_{0}^{−1}} ∘ \overline{f} ∘ \overline{h_{0}} = \overline{f}\overline{h_{0}}.

(3) It suffices to prove the first equality. ⊆: From (2) we conclude that for every F ∈ Autf_{L}(N^{∗}) there are h_{1}, …, h_{n} ∈ G^{∗} and f_{1}, …, f_{n} ∈ Autf_{L}(M^{∗}) such that F =

\overline{f_{1}}\overline{h_{1}} ∘ … ∘ \overline{f_{n}}\overline{h_{n}}. Then

F = \overline{h_{1}^{−1}f_{1}(h_{1})} ∘ \overline{f_{1}} ∘ \overline{h_{2}^{−1}f_{2}(h_{2})} ∘ \overline{f_{2}} ∘ … ∘ \overline{h_{n}^{−1}f_{n}(h_{n})} ∘ \overline{f_{n}}.

Using the rule \overline{f} ∘ \overline{g} = \overline{f(g)} ∘ \overline{f}, one can prove that F = \overline{g} ∘ \overline{f_{1} … f_{n}}, for some g ∈ G_{L} (for example \overline{f_{1}} ∘ \overline{h_{2}^{−1}f_{2}(h_{2})} = \overline{f_{1}(h_{2}^{−1})f_{1}(f_{2}(h_{2}))} ∘ \overline{f_{1}}, and f_{1}(h_{2}^{−1})f_{1}(f_{2}(h_{2})) = f_{1}(h_{2})^{−1}f_{2}^{f_{1}^{−1}}(f_{1}(h_{2})) ∈ X_{L}).

⊇: It is clear that Autf_{L}(N^{∗}) ⊇ Autf_{L}(M^{∗}) (use (2)). It is enough to prove that Autf_{L}(N^{∗}) ⊇ X_{L}. Assume that small tuples a, b satisfy b = f(a), for some f ∈ Autf_{L}(M^{∗}). We have to prove that \overline{a^{−1}b} ∈ Autf_{L}(N^{∗}). Since \overline{f}\overline{a} ∈ Autf_{L}(N^{∗}), we have \overline{a^{−1}b} = \overline{a^{−1}f(a)} = \overline{f}\overline{a} ∘ \overline{f^{−1}} ∈ Autf_{L}(N^{∗}). □

Now we characterize some invariant subgroups of G^{∗}: G_{∅}^{0}, G_{∅}^{00} and G_{∅}^{∞}, in terms of N^{∗}.

Proposition 3.4. (1) G_{L}^{∗} = G^{∗} ∩ Autf_{L}(N^{∗}) and G_{L}^{∗} is the smallest ∅-invariant

subgroup of G with bounded index in G^{∗} (i.e. G_{L}^{∗} = G_{∅}^{∞}).

(2) Let G′_{KP} = G^{∗} ∩ Autf_{KP}(N^{∗}), then G_{KP}^{∗} ⊆ G′_{KP} and G′_{KP} is the smallest ⋀- definable over ∅ subgroup with bounded index in G^{∗} (i.e. G′_{KP} = G_{∅}^{00}).

(3) Let G′_{Sh} = G^{∗} ∩ Autf_{Sh}(N^{∗}), then G_{Sh}^{∗} ⊆ G′_{Sh} and G′_{Sh} is the intersection of all ∅-definable subgroups of G^{∗} with finite index (i.e. G′_{Sh} = G_{∅}^{0}).

Proof. (1) The first equality follows directly from Proposition 3.3(3). Let H < G^{∗} be ∅-invariant with bounded index. It suffices to prove that X_{Θ} ⊆ H. Take an order inscernible sequence (a_{n})_{n<ω} (so Θ(a_{0}, a_{1})). If a_{0}^{−1}a_{1} ∉ H, then for every i < j < ω, a_{i}^{−1}a_{j} ∉ H, but we can extend an indiscernible sequence as much as we want, so the index [G^{∗} : H] is unboundedly large, a contradiction.

(2) If N′ ≺ N^{∗} is an arbitrary small model, then

G′_{KP} = \{g ∈ G^{∗}: E_{KP} (N′, \overline{g}(N′))\}.

Inclusion ⊆ is obvious. ⊇: If E_{KP} (N′, \overline{g}(N′)), then \overline{g}(N′) = F (N′) for some F ∈ Autf_{KP}(N^{∗}). Since \overline{g}|_{M^{∗}} = id_{M^{∗}}, F ∈ G^{∗}, and \overline{g}F^{−1} ∈ Autf_{L}(N^{∗}), so \overline{g} ∈ G′_{KP}.

G′_{KP} has bounded index (G_{L}^{∗} ⊆ G′_{KP}) and is ⋀-definable over N′x_{0}. In fact G′_{KP} is ∅-invariant. To see this let F = \overline{g′} ∘ \overline{f′} ∈ Aut(N^{∗}). Then

F [G′_{KP}] = \{F (g): E_{KP} (N′, \overline{g}(N′))\} = \{F (g): E_{KP} (F (N′), F (\overline{g}(N′)))\},

but F ∘ \overline{g} = \overline{g′} ∘ \overline{f′} ∘ \overline{g} = \overline{g′f′(g)g^{′−1}} ∘ F, thus

\begin{aligned}F [G′_{KP}] & = \{f′(g): E_{KP} (F (N′), \overline{g′} ∘ \overline{f′(g)g^{′−1}}(F (N′)))\} \\ & = \{f′(g): E_{KP} (\overline{g^{′−1}}F (N′), \overline{f′(g)}(\overline{g^{′−1}}F (N′))\} = G′_{KP} , \end{aligned}

and hence G′_{KP} is ⋀-definable over ∅. The relation E(x, y) = x^{−1}y ∈ G′_{KP} is bounded ⋀-definable over ∅, therefore E_{KP} |_{G^{∗}} ⊆ E and G_{KP}^{∗} ⊆ G′_{KP}. Take H < G^{∗}, another subgroup which is ⋀-definable over ∅ and has bounded index in G^{∗}. Then E_{KP} ⊆ E_{H}, so for g ∈ G′_{KP} we have E_{H}(x_{0}, \overline{g}(x_{0})) and then g^{−1} · x_{0} = \overline{g}(x_{0}) = \overline{h}(x_{0}) = h^{−1} · x_{0} for some h ∈ H. By regularity of · we obtain g = h ∈ H.

(3) As in (2) it can be proved that G′_{Sh} is ⋀-definable over ∅. Let g ∈ G′_{Sh}, and H < G^{∗} be a ∅-definable subgroup with finite index in G^{∗}. We show that g ∈ H. Consider the relation E(x, y) = (∃h ∈ H)(x = h · y) on X^{∗}. E is a ∅-invariant, finite equivalence relation on X^{∗}, thus E_{Sh}|_{X^{∗}} ⊆ E. By regularity of · we conclude that g ∈ H. If we consider E(x, y) = x^{−1}y ∈ H on G^{∗}, then E_{Sh}|_{G^{∗}} ⊆ E and therefore G_{Sh}^{∗} ⊆ H.

Let g belong to all ∅-definable subgroups of G^{∗} of finite index. We prove that \overline{g} ∈ Autf_{Sh}(N^{∗}). From Proposition 2.4 we know that Autf_{Sh}(N^{∗}) is the preimage under the quotient map j of the quasi-connected component QC of Gal_{L}(Th(N)). Let H ◁ Gal_{L}(Th(N)) be an open subgroup. It suffices to show that \overline{g} ∈ j^{−1}[H] ◁ Aut(N^{∗}). Note that the group H′ = j^{−1}[H] ∩ G^{∗} is ∅-invariant, because for f ∈ Aut(M^{∗}) if \overline{g} ∈ j^{−1}[H], then \overline{f(g)} = \overline{g}\overline{f^{−1}} ∈ j^{−1}[H]. H′ is also definable, because by Lemma 2.1(iv)(b), j^{−1}[H] = Aut(ℭ/\overline{m}/F)), so g ∈ H′ if and only if F (\overline{m}, \overline{g}(\overline{m}))). Hence H′ is a ∅-definable subgroup of G^{∗} of finite index and thus \overline{g} ∈ j^{−1}[H]. □

The compact topological group Gal_{L}(Th(N^{∗})) contains as a subgroup the group G^{∗}/G_{L}^{∗}, so we may ask about the induced topology on G^{∗}/G_{L}^{∗}. The next proposition describes this topology.

Proposition 3.5. (1) The induced subspace topology on G^{∗}/G_{L}^{∗} from Gal_{L}(Th(N)) is precisely the logic topology: let i: G^{∗} → G^{∗}/G_{L}^{∗} be the quotient map, then X ⊆ G^{∗}/G_{L}^{∗} is closed if and only if its preimage i^{−1}[X] ⊆ G^{∗} is ⋀-definable over some (equivalently every) small model. With this topology G^{∗}/G_{L}^{∗} is a compact topological group (this topology is Hausdorff if and only if G_{L}^{∗} is ⋀-definable).

(2) The topology of Gal_{L}(Th(M)) as the Lascar group of Th(M) and the induced topology on Gal_{L}(Th(M)) as a subspace of Gal_{L}(Th(N)) coincide.

(3) If X ⊆ G^{∗}/G_{L}^{∗} and Y ⊆ Gal_{L}(Th(M)) are closed, then X · Y ⊆ Gal_{L}(Th(N)) is also closed. In particular, if Th(M) is G-compact, then G^{∗}/G_{L}^{∗} is closed subgroup of Gal_{L}(Th(N)).

(4) The closure of identity in G^{∗}/G_{L}^{∗} is G′_{KP} /G_{L}^{∗}.

(5) The quasi-connected component (the intersection of all open subgroups) of G^{∗}/G_{L}^{∗} is G′_{Sh}/G_{L}^{∗}.

Proof. (1) Let N′ be a small model. Without loss of generality we may assume that x_{0} ∈ N′. The restriction of the quotient map j to G^{∗} is precisely the quotient map i.

We have the following commutative diagram:

\xymatrix{G^{∗} \ar@{->>}[r]^{i} \ar@{^{(}->}[d]^{⊆} & G^{∗}/G_{L}^{∗} \ar@{^{(}->}[d]_{⊆} \\ Aut(N^{∗}) \ar@{->>}[r]^{j} & Gal_{L}(Th(N))}

Let X ⊆ G^{∗}/G_{L}^{∗} be closed in the induced subspace topology, i.e. X = G^{∗}/G_{L}^{∗} ∩C, where C ⊆ Gal_{L}(Th(N)) is closed. Then ν^{−1}[C] is closed in S_{N′}(N′), so there exists a type Φ(x, y) over ∅ for which

ν^{−1}[C] = μ[j^{−1}[C]] = {tp(F (N′)/N′): F ∈ j^{−1}[C]} = {tp(N′′/N′): Φ(N′′, N′)}.

The subset i^{−1}[X] ⊆ G^{∗} is ⋀-definable over N′, because for g ∈ G^{∗}

g ∈ i^{−1}[X] ⇔ \overline{g} ∈ j^{−1}[C] ⇔ Φ(\overline{g}(N′), N′).

The implication ⇐ in the last equivalence holds, because if Φ(\overline{g}(N′), N′), then \overline{g}(N′) = F (N′) for some F ∈ j^{−1}[C], and thus j(\overline{g}) = j(F) ∈ C.

Now assume that i^{−1}[X] is ⋀-definable over N′, i.e. for g ∈ G^{∗}, g ∈ i^{−1}[X] if and only if Ψ(g, N′), for some type Ψ. Let C = X · Gal_{L}(Th(M)) ⊆ Gal_{L}(Th(N)). Then

X = G^{∗}/G_{L}^{∗} ∩ C.

In order to prove that C is closed in Gal_{L}(Th(N)) it is enough to show that

ν^{−1}[C] = {tp(F (N′)/N′): F ∈ j^{−1}[C]}
= {tp(N′′/N′): x_{0}^{N′′} = g^{−1} · x_{0}, Ψ(g, N′) holds and tp(N′′) = tp(N′)}.

The last equality holds because j^{−1}[C] = i^{−1}[X]∘Aut(M^{∗}), and if F = \overline{g}∘\overline{f}, g ∈ i^{−1}[X], then x_{0}^{N′′} = F (x_{0}) = \overline{g} ∘ \overline{f}(x_{0}) = g^{−1} · x_{0} (here N′′ = F (N′)).

(2) The proof is similar to the proof in (1) and we leave it to the reader.

(3) The set ν^{−1}[X · Y] = μ[j^{−1}[X · Y]] is closed in S_{N′}(N′) because it is equal to the following

\{tp(\overline{g} ∘ \overline{f}(N′)/N′): g ∈ i^{−1}[X], f ∈ j^{−1}[Y]\} = {tp(N′′/N′): tp(M′′/M′) ∈ ν^{−1}[Y], x_{0}^{N′′} = g^{−1} · x_{0}, g ∈ i^{−1}[X] and tp(N′′) = tp(N′)}.

Above we use the fact that j^{−1}[X · Y] = i^{−1}[X] ∘ ν^{−1}[Y].

(4) G′_{KP} /G_{L}^{∗} contains cl(id), because G′_{KP} is ⋀-definable over ∅. The subgroup i^{−1}[cl(id)] of G^{∗} is ⋀-definable over ∅ and of bounded index (because G_{L}^{∗} ⊆ i^{−1}[cl(id)]), thus G′_{KP} ⊆ i^{−1}[cl(id)].

(5) The group G′_{Sh} is the intersection of all ∅-definable subgroups of G^{∗} of finite index, thus G′_{Sh}/G_{L}^{∗} contains quasi-connected component QC (because if H < G^{∗} is ∅-definable of finite index, then H/G_{L}^{∗} is closed of finite index, hence open). Let H be an arbitrary open subgroup of G^{∗}/G_{L}^{∗}. It suffices to show that G′_{Sh}/G_{L}^{∗} ⊆ H. The group H is closed of finite index, hence H · Gal_{L}(Th(M)) is a closed subgroup of Gal_{L}(Th(N)) of finite index. Therefore

Autf_{Sh}(N^{∗}) ⊆ j^{−1}[H · Gal_{L}(Th(M))],

and then G′_{Sh} ⊆ i^{−1}[H]. This gives G′_{Sh}/G_{L}^{∗} ⊆ H. □

The next corollary motivates us to investigate ⋀-definability of G_{L}^{∗}. We do this in the next section. If G_{L}^{∗} is not ⋀-definable, then N may give us a new kind of not G-compact theory.

Corollary 3.6. Th(N) is G-compact if and only if Th(M) is G-compact and G_{L}^{∗} is ⋀-definable.

Proof. The topological group G is Hausdorff if nad only if {e_{G}} is closed and we can apply the previous proposition. □

Now we describe the relations Θ, E_{L}, E_{KP} and E_{Sh} on the sort X^{∗} in terms of orbits of the groups G_{L}^{∗}, G′_{KP} and G′_{Sh} from Proposition 3.3.

Lemma 3.7. Let x ∈ X^{∗} and n < ω.

(1) {y ∈ X^{∗}: Θ^{n}(x, y)} = X_{Θ}^{n} · x

(2) x/E_{L} = G_{L}^{∗} · x

(3) x/E_{KP} = G′_{KP} · x

(4) x/E_{Sh} = G′_{Sh} · x

Proof. (1) It is enough to prove this for n = 1. ⊆: Assume x, y ∈ X^{∗}, Θ(x, y) and y = g_{0}x for some g_{0} ∈ G^{∗}. We may assume that x = x_{0}. We can extend (x_{0}, g_{0}x_{0}) to an order indiscernible sequence (x_{0}, g_{0}x_{0}, g_{1}x_{0}, …) ⊆ X^{∗}. Then for 0 ≤ i_{1} < … < i_{n} < ω, 0 ≤ j_{1} < … < j_{n} < ω:

(x_{0}, g_{i_{1}}x_{0}, g_{i_{2}}x_{0}, …) ≡ (g_{j_{1}}x_{0}, g_{j_{2}}x_{0}, g_{j_{3}}x_{0}, …).

Applying the automorphism \overline{g_{j_{1}}} we obtain:

(g_{j_{1}}x_{0}, g_{j_{2}}x_{0}, g_{j_{3}}x_{0}, …) ≡ (x_{0}, g_{j_{2}}g_{j_{1}}^{−1}x_{0}, g_{j_{3}}g_{j_{1}}^{−1}x_{0}, …).

Hence from the previous two equivalences we get

(g_{i_{1}}x_{0}, g_{i_{2}}x_{0}, …) ≡_{x_{0}} (g_{j_{2}}g_{j_{1}}^{−1}x_{0}, g_{j_{3}}g_{j_{1}}^{−1}x_{0}, …),

so

(g_{i_{1}}, g_{i_{2}}, …) ≡ (g_{j_{2}}g_{j_{1}}^{−1}, g_{j_{3}}g_{j_{1}}^{−1}, …).

It means that (g_{0}, g_{1}, …) ⊆ G^{∗} is also order indiscernible and g_{0} ≡ g_{0}g_{1}^{−1}, so g_{0} ∈ X_{Θ}.

⊇: Let y = gx_{0} for g = ab^{−1} ∈ X_{Θ}, where Θ(a, b). We can find an indiscernible sequence (b, gb, …) ⊆ G^{∗}, and then (bx_{0}, gbx_{0}, …) ⊆ X^{∗} is also indiscernible, so Θ(bx_{0}, gbx_{0}). Applying \overline{b}, we obtain Θ(x_{0}, gx_{0}).

(2) Inclusion ⊇ follows from Proposition 3.3(3). ⊆: Let y = F (x) for some F = \overline{g}∘\overline{f} ∈ Autf_{L}(N^{∗}). We may assume that x = x_{0}. Then y = \overline{gf}(x_{0}) = \overline{g}x_{0} = g^{−1}x_{0} and g ∈ G_{L}.

(3) ⊇ follows from Proposition 3.4(2). Since E_{KP} |_{X^{∗}} ⊆ E_{G′_{KP}} |_{X^{∗}} we have ⊆.

(4) ⊇ follows from Proposition 3.4(3). ⊆: We know that G′_{Sh} = ⋂_{i∈I} H_{i}, where H_{i} is ∅-definable with finite index. Therefore E_{G′_{Sh}}|_{X^{∗}} = ⋂_{i∈I} E_{H_{i}}|_{X^{∗}}, so E_{Sh}|_{X^{∗}} ⊆ E_{G′_{Sh}}|_{X^{∗}} and we are done. □

Using Theorems 1.1 and 3.1 from [7] we can give a detailed analysis of Lascar and Kim-Pillay strong types on X^{∗}. This analysis describes also some basic properties of the group G. By diam(a) we denote the diameter of the Lascar strong type a/E_{L} (see [7]). Note that every two elements of X^{∗} have the same type over ∅, thus their Lascar strong types have the same diameter.

Remark 3.8. There are only two possibilities: Case 1 The diameters of all Lascar strong types on X^{∗} are infinite. The group G_{L}^{∗} is

not ⋀-definable, E_{L} ⫋ E_{KP}, G_{L}^{∗} ⫋ G′_{KP} (i.e. Th(N) is not G-compact) and

2^{ℵ_{0}} ≤ [G^{∗} : G_{L}^{∗}] = |X^{∗}/E_{L}| ≤ 2^{|T |}.

Case 2 There is n < ω such that for every x ∈ X^{∗}, diam(x) = n. Then E_{L}|_{X^{∗}} = E_{KP} |_{X^{∗}} = Θ^{n}|_{X^{∗}} and G_{L}^{∗} = X_{Θ}^{n} = G′_{KP} are ⋀-definable groups.

Lemma 3.9. (1) Either G′_{KP} = G_{L}^{∗}, or [G′_{KP} : G_{L}^{∗}] ≥ 2^{ℵ_{0}}.

(2) If the language of the structure M is countable, then either

G′_{Sh} = G′_{KP} or [G′_{Sh} : G′_{KP}] ≥ 2^{ℵ_{0}}.

In the last case the space of ∅-types S_{G}(∅) of G is of power 2^{ℵ_{0}}.

Proof. (1) follows from preceding remark, Lemma 3.7 and [7, Theorem 1.1].

(2) The proof is very similar to the proof of [4, Theorem 3.5], so we are brief. Consider the group H = G^{∗}/G′_{KP}. This group with the logic topology is a compact Hausdorff topological group. Since the language is countable, H is metrizable. Let d_{0} be a metric on H. Modifying d_{0} as in [4] we obtain an equivalent metric d, which is ∅-invariant. Since H is Hausdorff, the connected component of H is equal to the quasi-connected component QC, and by Proposition 3.5(5)

QC = G′_{Sh}/G′_{KP} .

Assume that G′_{Sh} ≠ G′_{KP} and take g ∈ G′_{Sh} \ G′_{KP}. Let r = d(e/G′_{KP} , g/G′_{KP}). For every δ with 0 < δ < r there is g_{δ} ∈ G′_{Sh} such that d(e/G′_{KP} , g_{δ}/G′_{KP}) = δ (because G′_{Sh}/G′_{KP} is connected). The metric d is ∅-invariant, hence for δ < δ′,

tp(g_{δ}) ≠ tp(g_{δ′}) and d(g_{δ′}/G′_{KP} , g_{δ}/G′_{KP}) ≥ δ′ − δ > 0.

Therefore the power of S_{G}(∅) is 2^{ℵ_{0}} and g_{δ′}g_{δ}^{−1} ∉ G′_{KP}, hence [G′_{Sh} : G′_{KP}] = 2^{ℵ_{0}}. □

4. ⋀-definability in G

In this section we investigate ⋀-definability of G_{L}^{∗} in several special cases.

Proposition 4.1. If the theory of M is small, then G_{L}^{∗} = G′_{KP} = G′_{Sh}. Hence G_{L}^{∗} is ⋀-definable.

Proof. Equality G_{L}^{∗} = G′_{KP} follows from [7, Theorem 3.1(2)]. Equality G′_{KP} = G′_{Sh} follows from Lemma 3.9. □

Proposition 4.2. If the theory of M is simple, then the theory of N is also simple and G_{L}^{∗} = X_{Θ}^{2} = G′_{KP}.

Proof. If Th(M) is simple, then Th(N) is also simple, because the structure N′ = (M, G, ·) (where ·: G × G → G is the group action) is definable in M. Thus N′ is simple, and N is obtained from N′ by forgetting some structure. Therefore Th(N) is also simple. In every simple structure E_{L} = E_{KP} = Θ^{2}, so G_{L}^{∗} = X_{Θ}^{2} follows from Lemma 3.7. □

Now we give a criterion for equality G′_{KP} = G′_{Sh}, when the theory of M is simple. If in this case G′_{KP} ⊊ G′_{Sh}, then it gives us a solution of an open problem: there exist an example of a structure with simple theory and in which Kim-Pillay and Shelah strong types are different (see Lemma 3.7). To state this criterion we need one definition. We call a subset P ⊆ G^{∗} thick if P is symmetric (P = P^{−1}) and there exist a natural number n < ω such that for any sequence g_{0}, …, g_{n−1} ∈ G there exist i < j < n such that

g_{i}^{−1} · g_{j} ∈ P.

When φ(x, y) is a thick formula (see Definition 1.1) then X_{φ} (see Definition 3.1) is thick set. On the other hand if P is definable thick set, then the formula φ_{P} (x, y) = x^{−1}·y ∈ P is also thick and P = X_{φ_{P}}. It is easy to see that for every n < ω we have

X_{Θ}^{n} = ⋂{X_{φ}^{n}: φ ∈ L is thick}.

Lemma 4.3. If M has a simple theory, then E_{KP} |_{X^{∗}} ⊊ E_{Sh}|_{X^{∗}} (i.e. G′_{KP} ⊊ G′_{Sh}) if and only if there exists a ∅-definable thick set P such that

G′_{Sh} ⊈ P^{2},

i.e. P^{2} does not contain any ∅-definable subgroup of G of finite index (see Proposition 3.4(3)).

Proof. If every thick P satisfies G′_{Sh} ⊆ P^{2}, then clearly G′_{Sh} ⊆ X_{Θ}^{2} = G_{L}^{∗} = G′_{KP}, so G′_{Sh} = G′_{KP}.

If G′_{Sh} = G′_{KP}, then from 4.2 we have that G′_{Sh} = X_{Θ}^{2} = ⋂{P^{2} : P is thick}. Thus every thick P satisfies G′_{Sh} ⊆ P^{2}. □

Example 4.4. There is an example of an abelian group (G, ·, …) which has a simple ω-categorical theory and satisfies X_{Θ} ⊊ X_{Θ}^{2} = G^{∗} (Example 6.1.10 in [2], private communication by E. Hrushovski). Consider a countable infinite dimensional vector space V over 𝔽_{2} = {0, 1}. Let B = {b_{i}: i < ω} be its basis and Q: V → 𝔽_{2} be the following degenerate orthogonal form with the induced scalar product (·, ·):

Q (∑_{i} λ_{i}b_{i}) = λ_{0}^{2} + λ_{1}λ_{2} + λ_{3}λ_{4} + …, (a, b) = Q(a + b) − Q(a) − Q(b), a, b ∈ V.

Q is degenerate, because its radical K = {v ∈ V : (v, ·) ≡ 0} = {0, b_{0}} is nontrivial. The structure G = (V, +, Q) has simple ω-categorical theory. We show that

X_{Θ} ⊆ V \ {b_{0}}.

If Θ(v, w), then Q(v) = Q(w). Assume on the contrary that v −w = b_{0}, then v = w+b_{0}, so:

Q(w) = Q(v) = Q(w + b_{0}) = Q(w) + Q(b_{0}) + (w, b_{0}) = Q(w) + 1,

and we reach a contradiction.

There are only 4 types over ∅: tp(0), tp(b_{0}), p(x), q(x), where p, q are types of elements v, w ≠ 0, b_{0} with Q(v) = 0, Q(w) = 1 respectively. The sets X_{Θ}, X_{Θ}^{2} are ∅- invariant, so they must be a union of some sets described by above types. Consider V_{0} = lin(b_{0}, b_{k}: k ≥ 5) ≺ V. It is easy to see that

(b_{1}, b_{4}) ≡_{b_{2}b_{3}V_{0}} (b_{1} + b_{3}, b_{4} + b_{2}), (b_{1}, b_{3}) ≡_{b_{2}b_{4}V_{0}} (b_{1} + b_{4}, b_{3} + b_{2}).

Thus by Lemma 1.2(ii) b_{3} = (b_{1}+b_{3})−b_{1} ∈ X_{Θ}^{2} and Q(b_{3}) = 0. Also b_{1} ≡_{V_{0}} b_{1}+b_{4}+b_{3}+b_{2},

so b_{4} + b_{3} + b_{2} ∈ X_{Θ}^{2} and Q(b_{4} + b_{3} + b_{2}) = 1. Therefore V \ {0, b_{0}} ⊆ X_{Θ}^{2} and then by Proposition 4.2 X_{Θ} ⊊ X_{Θ}^{2} = V.

The next proposition gives us ⋀-definability of G_{L}^{∗} for some special groups definable in the o-minimal theories.

Proposition 4.5. (1) If G is definably compact, definable in an o-minimal expansions of a real closed field, then G_{L}^{∗} = X_{Θ}^{2} = G′_{KP} = G^{00}.

(2) If (G, <, +, …) is an o-minimal expansion of an ordered group (G, <, +), then G^{∗} = G_{L}^{∗} = X_{Θ}^{2} = G^{00}.

Proof. (1) In [3] it is proved that under the above assumptions G has fsg and there exists G^{00} (the smallest definable subgroup of bounded index in G^{∗}). It is also proved that G^{00} is equal to

Stab(p) = {g ∈ G^{∗}: g · q = q},

for some (global) generic type p(x) ∈ S(G^{∗}). Since p is a type over the model G^{∗}, Stab(p) ⊆ X_{Θ}^{2}. Therefore

G^{00} = Stab(p) ⊆ X_{Θ}^{2} ⊆ G_{L}^{∗} ⊆ G′_{KP} = G_{∅}^{00} = G^{00}.

(2) By [8, Corollary 2.6] we can find a global type p(x) ∈ S(G^{∗}), satisfying Stab(p) = G^{∗}. Therefore G^{∗} = G_{L}^{∗} = X_{Θ}^{2} = G^{00}. □

Case 1 from Remark 3.8 may lead us to a new example of a non-G-compact theory. There is a criterion for ⋀-definability of G_{L}^{∗} [7, Theorem 3.1]: G_{L}^{∗} is ⋀-definable if

and only if G_{L}^{∗} = X_{Θ}^{n} for some n < ω. Thus if X_{Θ} generates a group in infinitely many steps, then G_{L}^{∗} is not ⋀-definable and Case 1 holds.

We have some further partial results concerning ⋀-definability of G_{L}^{∗}. These results involve generic subsets of G and measures on G. They will be a part of Ph.D. thesis of the first author and appear in a forthcoming paper.

5. More examples

We were not able to construct an example of a group G, where G_{L} is not ⋀-definable. We can try at least to construct a group G, where G_{E} is not ⋀-definable for some equivalence relation E other that E_{L} (which gives rise to G_{L}).

It is rather easy to find such examples even in the stable case, with the relation E ⋀-definable and coarser than equality of types ≡.

However even in the stable case we were not able to construct an example of G where G_{≡} is not ⋀-definable, although we conjecture such an example exists. In this case G_{Sh}^{∗} equals G^{0}, and is type definable, and equals G_{L}.

Since we are interested in finding an example where G_{L} is not ⋀-definable, naturally we are interested in non-⋀-definable G_{E}, where E is close to E_{L}.

In this section we give only an example (Example 5.1), where G_{≡} is not ⋀-definable. We could not come closer to E_{L} than ≡. We give also an example (Example 5.2) of a group G with non-G-compact theory.

Example 5.1. In [6] there is an example (for every n < ω) of a finite group G_{n} in which commutators X_{Inn(G_{n})} generate commutant G′_{n} = [G_{n}, G_{n}] = G_{Inn(G_{n})} in precisely n steps. We expand the structure (G_{n}, ·) to obtain a structure G_{n} satisfying

Aut(G_{n}) = Inn(G_{n}),

i.e. every automorphism of G_{n} is an inner automorphism of G_{n}. Note that in G_{n} the set X_{≡} equals X_{Inn(G_{n})} and generates a group in n steps. Consider the product ∏_{n<ω} G_{n} of the groups G_{n}. We expand ∏_{n<ω} G_{n} to a structure G as follows. For each k let E_{k} be the equivalence relation on ∏_{n<ω} G_{n} given by

E_{k}(u, v) ⇔ u(k) = v(k).

Then ∏_{n<ω} G_{n}/E_{k} is naturally identified with G_{k}. We expand ∏_{n<ω} G_{n} by the relations E_{k}, k < ω, and the G_{k}-structure on G_{k} (identified with ∏_{n<ω} G_{n}/E_{k}). We denote the quotient map

∏_{n<ω} G_{n} → ∏_{n<ω} G_{n}/E_{k}

by π_{k}.

Let G^{∗} be a large saturated extension of G. We will prove that in G^{∗}, the group G_{≡}^{∗} is not ⋀-definable. This boils down to proving that

(∗)  π_{k}[X_{≡}^{G^{∗}}] = X_{≡}^{G_{k}}.

Indeed, suppose the above holds. Then, since X_{≡}^{G_{k}} generates a group in ≥ n steps, also X_{≡}^{G^{∗}} generates a group in ≥ n steps. As n is arbitrary, we get that X_{≡}^{G^{∗}} generates the group G_{≡}^{∗} in infinitely many steps. By [7, Theorem 3.1(1)], the group G_{≡}^{∗} is not ⋀-definable.

Now we prove (∗). ⊆ is clear, since every automorphism of G^{∗} induces an automorphism of G_{k}. To prove ⊇, consider a, b ∈ G_{k} with b = f(a) for some f ∈ Aut(G_{k}). We can extend f to an automorphism of G and then to G^{∗}. If a = π_{k}(a′) for a′ ∈ G^{∗}, then

b = f(a) = π_{k}(f(a′)),

and therefore π_{k}(a^{′−1}f(a′)) = a^{−1}b.

Now we give an example of group G whose theory is not G-compact, but case 2 from Remark 3.8 holds.

Example 5.2. First we construct a group with a large finite diameter of Lascar strong types. Let M_{0} = (M_{0}, R, f) be a dense circular ordering (with respect to a ternary relation R), equipped with a function f, which is a cyclic bijection of M_{0} respecting R, of period 3. This structure was considered in [1] to construct the first example of a non-G-compact theory. Our group M_{3} will be the disjoint union of M_{0} and {0} equipped with a structure of vector space over 𝔽_{2} (i.e. M_{3} will be an abelian group of exponent 2) so that the addition + on M_{3} is “independent” of f and R.

To be more specific, let L be the language consisting of a ternary relation symbol R, function symbols + (binary) and f (unary) and a constant 0. To express “independence” of f and + we define inductively a set of terms T in L as follows.

Definition 5.3. Let T be the smallest set of terms of L such that:

(1) v, f(v), f^{2}(v) are in T for every variable v.

(2) If τ_{1}, …, τ_{k} (k ≥ 2) are distinct terms in T , then the terms f(τ_{1} + … + τ_{k}) and

f^{2}(τ_{1} + … + τ_{k}) are in T . Let T (\overline{x}) be the set of terms in T in variables \overline{x}.

+ will be interpreted as an associative operation, so we may omit parentheses in τ_{1} + … + τ_{k} in condition (2) in Definition 5.3. f will be interpreted as a cyclic function of period 3, so in Definition 5.3 there is no need to consider f^{k} for k ≥ 3.

Definition 5.4. Let C be the class of L-structures (V, +, 0, R, f) such that: (1) (V, +, 0) is a vector space over 𝔽_{2}, of infinite dimension.

(2) R is a circular order on the set V^{∗} = V \ {0}.

(3) f is a cyclic bijection of V^{∗} of period 3, respecting R. That is, every point of V^{∗} is a cyclic point of f of period 3 and R(x, y, z) implies R(f(x), f(y), f(z)). Also, R(x, f(x), f^{2}(x)) holds in V^{∗}.

(4) For every a ∈ V^{∗}, the set T (a) = {τ(a): τ ∈ T } is lineary independent.

Condition (4) in Definition 5.4 expresses the fact that in the structures in C f is “independent” of the vector space structure.

C is an elementary class with the joint embedding and amalgamation properties. Let T_{3} be the model completion of Th(C). T_{3} has quantifier elimination and its models are the existentially closed structures in C. We describe the 1-types in T_{3}.

Let ℭ be a monster model of T_{3}. Let a ∈ ℭ^{∗} = ℭ \ {0}. The type of a is determined by the way in wich the linear span lin(T (a)) is circularly ordered by R, or even by the way in which the set lin(T (a)) ∩ (a, f(a)) is linearly ordered by R. Here for a ≠ b ∈ ℭ^{∗}

(a, b) = {c ∈ ℭ^{∗}: R(a, c, b)}, [a, b) = {a} ∪ (a, b).

R induces on (a, b) a linear ordering that we denote by <. So there are 2^{ℵ_{0}} complete 1-types over ∅ in T_{3}.

We say a few words about indiscernible sequences in ℭ. First, if (a_{n})_{n<ω} is an infinite indiscernible sequence in ℭ, then a_{1} ∈ (a_{0}, f(a_{0})) or a_{1} ∈ (f^{2}(a_{0}), a_{0}).

Secondly, we point how to construct an indiscernible sequence in ℭ. Assume p(x) = tp(a) for some a ∈ ℭ^{∗}. Let C^{−}(a), C^{+}(a) be a Dedekind cut in the set lin(T (a)) ∩ (a, f(a)). That is, C^{−}(a) < C^{+}(a) and C^{−}(a) ∪ C^{+}(a) = lin(T (a)) ∩ (a, f(a)).

It follows that for every a′ realising p, the corresponding sets C^{−}(a′), C^{+}(a′) are a Dedekind cut in the set lin(T (a′)) ∩ (a′, f(a′)) and also the sets f(C^{−}(a′)), f(C^{+}(a′)) and f^{2}(C^{−}(a′)), f^{2}(C^{+}(a′)) are Dedekind cuts in the sets lin(T (a′))∩(f(a′), f^{2}(a′)) and lin(T (a′)) ∩ (f^{2}(a′), a′), respectively.

We can find a sequence (a_{n})_{n<ω} of elements of ℭ^{∗} such that for every n > m,

[a_{n}, f(a_{n})) ⊆ (a_{m}, f(a_{m})) and C^{−}(a_{m}) < [a_{n}, f(a_{n})) < C^{+}(a_{m}).

Using the Ramsey theorem we can find such a sequence that is moreover indiscernible. Using indiscernible sequences like that we see that p(x) is a strong Lascar type of diameter at least 3 and at most 6.

Similarly, replacing period 3 by period n (n ≥ 3), we construct a G-compact group M_{n} = (M_{n}, +, 0, R_{n}, f_{n}) with the diameter of Lascar strong types ≥ n.

Now let M = ∏_{3≤n<ω} M_{n} be the product of the groups M_{n}. Let E_{n} be the equivalence relation on M given by E_{n}(x, y) ⇔ x(n) = y(n). Then M/E_{n} is naturally identified with M_{n}.

We consider M as a group expanded by the relations E_{n} and the relations R_{n} and functions f_{n} on M/E_{n}. Hence in M there is no finite bound on the diameter of Lascar strong types. By [7], M is not G-compact.

References

[1] E. Casanovas, D. Lascar, A. Pillay, M. Ziegler, Galois groups of first order theories. J. Math. Log. 1 (2001), no. 2, 305–319

[2] G. Cherlin, E. Hrushovski, Finite structures with few types. Annals of Mathematics Studies, 152. Princeton, NJ, 2003.

[3] E. Hrushovski, Y. Peterzil, A. Pillay, Groups, measures, and the NIP. To appear in Journal of the AMS.

[4] K. Krupiński, L. Newelski, On Bounded Type-Definable Equivalence Relations. Notre Dame Journal of Formal Logic 43 (2002) no. 4, 231–242

[5] D. Lascar, A. Pillay, Hyperimaginaries and automorphism groups. J. Symbolic Logic 66 (2001), no. 1, 127–143

[6] I. D. Macdonald, On cyclic commutator subgroups. J. London Math. Soc. 38 (1963) 419–422 [7] L. Newelski, The diameter of a Lascar strong type. Fundam. Math. 176 (2003), no. 2, 157–170 [8] M. Petrykowski, Weak generic types and coverings of groups II. preprint 2007

[9] M. Ziegler, Introduction to the Lascar group. London Math. Soc. Lecture Note Ser., 291, 279–298

Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland

E-mail address: gismat@math.uni.wroc.pl, newelski@math.uni.wroc.pl