THE RELATIVIZED LASCAR GROUPS, TYPE-AMALGAMATION, AND ALGEBRAICITY

0. Introduction and background

Given a complete theory T, the notion of the Lascar (Galois) group $\operatorname {Gal}_{\operatorname {L}}(T)$ was introduced by D. Lascar (see [Reference Casanovas, Lascar, Pillay and Ziegler2]). The Lascar group only depends on the theory and it is a quasi-compact topological group with respect to a quotient topology of a certain Stone type space over a model ([Reference Casanovas, Lascar, Pillay and Ziegler2] or [Reference Ziegler and Tent13]). More recently, the notions of the relativized Lascar groups were introduced in [Reference Dobrowolski, Kim and Lee3] (and studied also in [Reference Krupiński, Newelski and Simon11] in the context of topological dynamics). Namely, given a type-definable set X in a large saturated model of the theory T, we consider the group of automorphisms restricted to the set X quotiented by the group of restricted automorphisms fixing the Lascar types of the sequences from X of length $\lambda $ . The relativized Lascar groups also only depend on T, on the type that defines X, and on the cardinal $\lambda $ . These groups are endowed with the quasi-compact quotient topologies induced by the canonical surjective maps from $\operatorname {Gal}_{\operatorname {L}}(T)$ to the relativized Lascar groups.

This paper continues the study started in [Reference Dobrowolski, Kim and Lee3], where a connection between the relativized Lascar groups of a strong type and the first homology group of the strong type was established. If T is G-compact (for example when T is simple), then the relativized Lascar group of a strong type is compact and connected, and we use compact group theory to obtain results presented here.

A common theme for the results in this paper is the connection between group theoretic properties of the Lascar group of a strong type and the model theoretic properties of the type. In Section 1, we use the fact that any compact connected group is divisible to show that any strong type p in a simple theory has a property we call divisible amalgamation. The property would follow from the Independence Theorem if p was a Lascar type, but here p is only assumed to be a strong type. Moreover, we give amalgamation criteria for when the Lascar group of the strong type of a model is abelian, and for when the strong type of a model is a Lascar type.

In Section 2, we study morphisms between the relativized Lascar groups. The main goal is to understand the connection between the Lascar group of $\operatorname {stp}(a)$ , for some tuple a, and the Lascar group of $\operatorname {stp}(\operatorname {acl}(a))$ or $\operatorname {stp}(ac)$ , for a finite tuple $c\in \operatorname {acl}(a)$ . We prove that if T is G-compact, then for a finite tuple c algebraic over a, the restriction map from a Lascar group of $\operatorname {stp}(ac)$ to that of $\operatorname {stp}(a)$ is a covering map. In addition, if the Lascar group of $\operatorname {stp}(ac)$ is abelian then this group is isomorphic to that of $\operatorname {stp}(a)$ as topological groups. In order to achieve this, we separately prove a purely compact group theoretical result that any compact connected abelian group is isomorphic to its quotient by a finite subgroup. We also give an example showing that the abelianity of the relativized Lascar group is essential in the isomorphism result.

In Section 3, we mainly present three counterexamples: a non-G-compact theory where $\operatorname {stp}(a)$ is a Lascar type but $\operatorname {stp}(\operatorname {acl}(a))$ is not a Lascar type; an example showing that in the above-mentioned isomorphism result in Section 2, the tuple c being finite is essential; and an example answering a question raised in [Reference Kim and Lee10], namely a Lie group structure example where an RN-pattern minimal 2-chain is not equivalent to a Lascar pattern 2-chain having the same boundary.

Let us remark that the results of Section 2 and the supplementary examples from Sections 3.1 and 3.2 do not make any use of the homology groups $H_1(p)$ , so a reader interested only in these results is advised to skip from Fact 0.4 to Section 2.

In the remaining part of this section we recall the definitions and terminology of basic notions, which, unless said otherwise, we use throughout this note. We work in a large saturated model ${\mathcal M}(={\mathcal M}^{\operatorname {eq}})$ of a complete theory T, and we use the standard notation. So $A, B,\dots $ and $M,N,\dots $ are small subsets and elementary submodels of ${\mathcal M}$ , respectively. Lower-case letters $a,b,\dots $ will denote tuples of elements from ${\mathcal M}$ , possibly infinite. We will explicitly specify when a tuple is assumed to be finite.

To simplify the notation, we state the results for types over the empty set (rather than over $\operatorname {acl}(\emptyset )$ ). This does not reduce the generality because, after naming a parameter set, we assume for the rest that $\operatorname {dcl}(\emptyset )(=\operatorname {dcl}^{\operatorname {eq}}(\emptyset ))=\operatorname {acl}(\emptyset )(=\operatorname {acl}^{\operatorname {eq}}(\emptyset ))$ . We fix a complete strong type $p(x)\in S(\emptyset )$ with possibly infinite arity of x. For tuples $a,b$ , we write $a\equiv _Ab$ ( $a\equiv ^s_Ab$ , resp.) to mean that they have the same (strong, resp.) type over A. Note that $a\equiv ^s_Ab$ if and only if $a\equiv _{\operatorname {acl}(A)}b$ .

Let us recall the definitions of the Lascar groups and types. These are well-known notions in model theory. In particular, the Lascar group depends only on T and is a quasi-compact group under the topology introduced in [Reference Casanovas, Lascar, Pillay and Ziegler2] or [Reference Ziegler and Tent13]. Moreover, due to our assumption $\operatorname {dcl}(\emptyset )=\operatorname {acl}(\emptyset )$ , the group is connected as well. Recall that a topological space is compact if it is quasi-compact and Hausdorff.

Note that the above is the definition of “G-compactness over $\emptyset $ ,” but for convenience throughout this paper we omit “over $\emptyset $ .”

Now let us recall, mainly from [Reference Dobrowolski, Kim and Lee3], the definitions of various relativized Lascar groups of p and related facts.

We will give an example (in Example 2.3) where $\operatorname {Gal}_{\operatorname {L}}^1(p)$ and $\operatorname {Gal}_{\operatorname {L}}^2(p)$ are distinct. In [Reference Dobrowolski, Kim and Lee3, Remark 3.4], a canonical topology on each of the above groups was defined. With these topologies, they become quotients of the topological group $\operatorname {Gal}_{\operatorname {L}}(T)$ .

The last statement in the above fact explains the connection between the relativized Lascar group and the model theoretic homology group $H_1$ of the type. Let us recall now the key definitions in the homology theory in model theory. We fix a ternary automorphism-invariant relation between small sets of ${\mathcal M}$ satisfying

• • finite character: for any sets $A,B,$ and C, we have iff for any finite tuples $a\in A$ and $b\in B$ ;

• • normality: for any sets A, B, and C, if , then ;

• • symmetry: for any sets $A,B,$ and C, we have iff ;

• • transitivity: iff and , for any sets A and $B\subseteq C\subseteq D$ ;

• • extension: for any sets A and $B\subseteq C$ , there is $A'\equiv _BA$ such that .

Throughout this paper we call the above axioms the basic 5 axioms. We say that A is $*$ -independent from B over C if . Notice that there is at least one such relation for any theory, namely, the trivial independence relation given by: For any sets $A,B,C$ , put . Of course there is a non-trivial such relation when T is simple or rosy, given by forking or thorn-forking, respectively.

In the rest we may call a $*$ -independent n-simplex in p just an n-simplex of p, as far as no confusion arises.

Definition 0.7. Let $n\geq 0$ . We define:

$$ \begin{align*} &S_n({\mathcal A}^*_p):= \{\, f\in {\mathcal A}^*_p \mid f\ \mbox{is an}\ n\mbox{-simplex of}\ p\, \},\\ &C_n({\mathcal A}^*_p): = \mbox{the free abelian group generated by}\ S_n({\mathcal A}^*_p). \end{align*} $$

An element of $C_n({\mathcal A}^*_p)$ is called an n-chain of p. The support of a chain c, denoted by $\operatorname {supp}(c)$ , is the union of the supports of all the simplices that appear in c with a non-zero coefficient. Now for $n\geq 1$ and each $i=0, \ldots , n$ , we define a group homomorphism

$$ \begin{align*} \partial^i_n\colon C_n({\mathcal A}^*_p) \rightarrow C_{n-1}({\mathcal A}^*_p) \end{align*} $$

by putting, for any n-simplex $f\colon {\mathcal P}(s) \rightarrow {\mathcal C}$ in $S_n({\mathcal A}^*_p)$ where $s = \{ s_0 <\cdots < s_n \}\subseteq \omega $ ,

$$ \begin{align*} \partial_n^i(f):= f\restriction {\mathcal P}(s\backslash\{s_i\}) \end{align*} $$

and then extending linearly to all n-chains in $C_n({\mathcal A}^*_p)$ . Then we define the boundary map

$$ \begin{align*} \partial_n\colon C_n({\mathcal A}^*_p)\rightarrow C_{n-1}({\mathcal A}^*_p) \end{align*} $$

by

$$ \begin{align*} \partial_n(c):=\sum_{0\leq i\leq n}(-1)^i \partial_n^i(c). \end{align*} $$

We shall often refer to $\partial _n(c)$ as the boundary of c. Next, we define:

$$ \begin{align*} &Z_n({\mathcal A}^*_p):= \operatorname{Ker} \, \partial_n,\\ & B_n({\mathcal A}^*_p):= \operatorname{Im} \, \partial_{n+1}. \end{align*} $$

The elements of $Z_n({\mathcal A}^*_p)$ and $B_n({\mathcal A}^*_p)$ are called n-cycles and n-boundaries in p, respectively. It is straightforward to check that $ \partial _{n}\circ \partial _{n+1} =0.$ Hence we can now define the group

$$ \begin{align*} H^*_n(p):= Z_n({\mathcal A}^*_p)/B_n({\mathcal A}^*_p) \end{align*} $$

called the nth $*$ -homology group of p.

Definition 0.9. A $1$ -chain $c\in C_1({\mathcal A}^*_p)$ is called a $1$ - $*$ -shell (or just a $1$ -shell) in p if it is of the form

$$ \begin{align*}c =f_0-f_1+f_2, \end{align*} $$

where $f_i$ ’s are $1$ -simplices of p satisfying

$$ \begin{align*} \partial^i f_j = \partial^{j-1} f_i \quad \mbox{whenever}\ 0\leq i < j \leq 2. \end{align*} $$

Hence, for $\operatorname {supp}(c)=\{n_0< n_1< n_2\}$ and $ k\in \{0,1, 2\}$ , it follows that

$$ \begin{align*}\operatorname{supp}(f_k)=\operatorname{supp}(c)\backslash \{n_k\}.\end{align*} $$

Notice that the boundary of any $2$ -simplex is a $1$ -shell. Recall that a notion of an amenable collection of functors into a category is introduced in [Reference Goodrick, Kim, Kolesnikov, Jang, Kim, Lee and Yie5]. Due to the 5 axioms of , it easily follows that ${\mathcal A}^*_p$ forms such a collection of functors into ${\mathcal C}$ . Hence the following corresponding fact holds.

Fact 0.10 [Reference Goodrick, Kim, Kolesnikov, Jang, Kim, Lee and Yie5] or [Reference Dobrowolski, Kim and Lee3]

$$ \begin{align*} H^*_1(p) = \{ [c] \mid c \mbox{ is a } 1-\mbox{*}\mbox{-shell with}\ \operatorname{supp}(c) = \{ 0, 1, 2 \} \, \}. \end{align*} $$

We now recall basic notions and results that appeared in [Reference Dobrowolski, Kim and Lee3]. For the rest of this section we assume that p is the strong type of an algebraically closed set.

We summarize some properties of endpoint pairs of $1$ -shells. We define an equivalence relation $\sim ^*$ on the set of pairs of realizations p as follows: For $a,a',b,b'\models p$ , $(a,b)\sim ^*(a',b')$ if two pairs $(a,b)$ and $(a',b')$ are endpoint pairs of $1$ -shells s and $s'$ respectively such that $[s]=[s']\in H^*_1(p)$ . We write ${\mathcal E}^*=p({\mathcal M})\times p({\mathcal M})/\sim ^*$ . We denote the class of $(a,b)\in p({\mathcal M})\times p({\mathcal M})$ by $[a,b]$ . Now, define a binary operation $+_{{\mathcal E}^*}$ on ${\mathcal E}^*$ as follows: For $[a,b],[b',c']\in {\mathcal E}^*$ , $[a,b]+_{{\mathcal E}^*} [b',c']=[a,c]$ , where $bc\equiv b'c'$ .

Remark 0.15. (1) Assume that $p(x)$ is the type of a small model. Then for any $M, N\models p$ , the equivalence classes of (equality of) Lascar types of M and N are interdefinable: Let $M\equiv ^L M'$ and $MN\equiv M'N'$ . It suffices to show that $N\equiv ^L N'$ . Now there is $f\in \operatorname {Autf}({\mathcal M})$ such that $f(M')=M$ . Let $N'':=f(N')$ so that $N''\equiv ^LN'$ and $MN\equiv MN''$ . Hence $N\equiv _{M}N''$ and $N\equiv ^L N''\equiv ^L N'$ as wanted. When p is a type of any tuple we will see that the same holds if $\operatorname {Gal}_{\operatorname {L}}^1(p)$ is abelian (Remark 1.10).

(2) Notice that for tuples $a,b\models p$ , we have:

• (**) there is a commutator f in $\operatorname {Aut}(p)$ such that $f(a)=b$ if and only if there are $d^1,d^2,d^3\models p$ such that $ad^1\equiv d^3d^2$ and $bd^3\equiv d^1d^2$ .

Now we recall the following fact of compact group theory by M. Gotô from [Reference Hofmann and Morris6, Theorem 9.2]: Assume $(F,\cdot )$ is a compact connected topological group. Then $F'$ , the commutator subgroup of F, is simply the set of commutators in F, i.e.,

$$ \begin{align*}F'=\{ f\cdot g \cdot f^{-1}\cdot g^{-1}\mid f,g \in F\}.\end{align*} $$

It follows that $F'$ is a closed subgroup of F, and both $F'$ and $F/F'$ are compact connected groups as well.

Due to the theorem we newly observe here that if T is G-compact then in Fact 0.14(1)(c), we can choose $n=1$ : If the equivalent conditions in Fact 0.14(1) hold, then we have that $h(a)=b$ for some h in the commutator subgroup of $\operatorname {Aut}(p)$ . Now, as T is G-compact, we get by Gotô’s Theorem that h is a commutator in $\operatorname {Aut}(p)$ , so, by $(**)$ , there are $d^1,d^2,d^3\models p$ such that $bd^3\equiv d^1d^2$ .

1. Amalgamation properties of strong types in simple theories

Note that if T is simple then since T is G-compact, each $\operatorname {Gal}_{\operatorname {L}}^{\lambda }(p)$ is a compact (i.e., quasi-compact and Hausdorff) connected group, so it is divisible (see [Reference Hofmann and Morris6, Theorem 9.35]). In this section we assume T is simple (except Remark 1.10 and Example 1.11), and the independence is nonforking independence. It is still an open question whether the strong type p is necessarily a Lascar type. If so, then the following theorem follows easily by the $3$ -amalgamation of Lascar types in simple theories. But regardless of the answer to the question, p has the following amalgamation property. We do not assume here that a realization of p is algebraically closed.

Now we assume that p is the type of an algebraically closed tuple. For $a,a'\models p$ , we have a better description of $[a,a']=0$ in $H_1(p).$

Proof (1) $\Rightarrow $ (2) Since T is G-compact, by Remark 0.15(2) there are $b',c',d'\models p$ such that $ab'\equiv c'd'$ and $b'd'\equiv a'c'$ . Now there is $b\equiv ^L b'$ such that . Hence, by $3$ -amalgamation, there is $d_0\equiv ^L d'$ such that $d_0\models \operatorname {tp}(d'/b) \cup \operatorname {tp}(d''/b')$ and , where $d''b'\equiv d'b$ and $d''\equiv ^L d'\equiv ^L d_0$ . By extension there is no harm in assuming that .

Now since $a'c'\equiv b'd'$ , there are $a''c$ such that $b'd'bd_0\equiv a'c'a''c$ . Hence, $c\equiv ^L c'$ , $a''\equiv ^L a'$ , and . Again by extension, we can further assume that . Moreover, since $ab'\equiv c'd'$ , there is $d_1\equiv ^L d'$ such that $ab'b\equiv c'd'd_1$ . Then since with $c\equiv ^Lc$ , and , again by $3$ -amalgamation we can assume that $c'd_1\equiv cd_1$ and .

Now the situation is that , , and . Moreover, $d_0\equiv ^L d'\equiv ^L d_1$ . Hence, by $3$ -amalgamation, we have $d\models \operatorname {tp}(d_0/aa'b)\cup \operatorname {tp}(d_1/c)$ and . Therefore, each of $\{a,b,c,d\}$ , $\{a', b,c,d\}$ is independent. Moreover, due to above combinations

$$ \begin{align*}ab\equiv c'd_1\equiv cd_1\equiv cd,\end{align*} $$

and

$$ \begin{align*}bd\equiv bd_0\equiv bd'\equiv b'd''\equiv b'd_0\equiv a'c\end{align*} $$

as wanted.

(2) $\Rightarrow $ (1) Clear by Remark 0.15(2).⊣

From now until Theorem 1.8, we assume that $p(x)$ is the ( strong ) type of a small model, and all tuples $a,b,c,\dots $ realize p, so all are universes of models. Hence, by Fact 0.4, $\operatorname {Autf}^1(p)=\operatorname {Autf}_{\operatorname {fix}}(p)$ and $(\operatorname {Gal}_{\operatorname {L}}(T)\cong )\operatorname {Gal}_{\operatorname {L}}^{\operatorname {fix}}(p)=\operatorname {Gal}_{\operatorname {L}}^{1}(p)$ , which we simply write $\operatorname {Autf}(p)$ and $\operatorname {Gal}_{\operatorname {L}}(p)$ , respectively. Moreover, $H_1(p)\cong \operatorname {Gal}_{\operatorname {L}}(p)/(\operatorname {Gal}_{\operatorname {L}}(p))'$ and hence it is a compact connected (so divisible) abelian group.

Proof (1) $\Rightarrow $ (2) By (1), there are $b_0, d_0\models p$ such that $\{a,c,b_0,d_0\}$ is independent, $ab_0,cd_0\models r$ , and $ac, b_0d_0\models s$ . Hence, there is $d_1$ such that $abd_1\equiv ab_0d_0$ . Now by $3$ -amalgamation over the model a, there is

$$ \begin{align*}d\models \operatorname{tp}(d_0/a;c) \cup \operatorname{tp}(d_1/a;b),\end{align*} $$

such that $\{a,b,c,d\}$ is independent. Moreover, $cd\equiv cd_0\models r$ and $bd\equiv bd_1\equiv b_0d_0 \models s$ , as desired.

(2) $\Rightarrow $ (1) Clear.⊣

By the same proof we obtain the following too.

Proof (1) $\Rightarrow $ (2) Assume (1). Thus for $G:=\operatorname {Aut}(p)$ , we have $G'\leq \operatorname {Autf}(p)$ . Now let $r(x,y)$ and $s(x,y)$ be some complete types containing . There are independent $a,a', b,d,c\models p$ such that $ab,cd\models r$ and $bd, a'c\models s$ . Hence there is a commutator $f\in G'\leq \operatorname {Autf}(p)$ such that $f(a)=a'$ , so $a\equiv ^L a'$ . Then by $3$ -amalgamation of $\operatorname {Lstp}(a)$ , there is

$$ \begin{align*}a_0\models \operatorname{tp}(a/bd)\cup \operatorname{tp}(a'/c)\end{align*} $$

such that $\{a_0,b,c,d\}$ is independent, $a_0b\equiv ab\models r$ , and $a_0c\equiv a'c\models s$ . Therefore p has commutative amalgamation.

(2) $\Rightarrow $ (1) To show (1), due to Fact 0.14(2) it is enough to prove that if $a,a'\models p$ and $[a,a']=0\in H_1(p)$ ( $\dagger $ ), then $a\equiv ^L a'$ . Now assume (2) and ( $\dagger $ ). Thus, by Proposition 1.2, there are $b,c,d\models p$ such that each of the sets $\{a,b,c,d\}$ and $\{a', b,c,d\}$ is independent, $ab\equiv cd$ , and $bd\equiv a'c$ . Now by Lemma 1.4 and extension there is $c'\models p$ such that and $ac'\equiv bd\equiv a'c$ and $c'd\equiv ab\equiv cd$ . Then, since d is a model, we have $c \equiv ^L c'$ , and there is $a''$ such that $a'c\equiv _da''c'$ , so $a\equiv ^L a''$ and $a''\equiv _{c'}a$ . Since $c'$ is a model as well, we conclude that $a'\equiv ^L a''\equiv ^L a$ .⊣

A proof similar to that of the above Theorem 1.6 (2) $\Rightarrow $ (1) applying Fact 0.14(1)(b) gives the following as well.

A stronger consequence is obtained.

2. Relativized Lascar Galois groups and algebraicity

As mentioned in the introduction, this section does not make any use of the homology groups and thus can be read independently from Section 1. Recall that throughout we assume that $\operatorname {acl}(\emptyset )=\operatorname {dcl}(\emptyset )$ . In Remark 1.8 from [Reference Dobrowolski, Kim and Lee3] it was noticed that if $\operatorname {tp}(a)$ is a Lascar type, and a finite tuple c is algebraic over a then $\operatorname {tp}(ac)$ is also a Lascar type, and hence if additionally T is G-compact then $\operatorname {tp}(\operatorname {acl}(a))$ is also a Lascar type. In other words, if $\operatorname {Gal}_{\operatorname {L}}^1(\operatorname {tp}(a))$ is trivial, then $\operatorname {Gal}_{\operatorname {L}}^1(\operatorname {tp}(\operatorname {acl}(a))$ is trivial when T is G-compact. It seems natural to ask whether, more generally, $\operatorname {Gal}_{\operatorname {L}}^1(\operatorname {tp}(a))$ and $\operatorname {Gal}_{\operatorname {L}}^1(\operatorname {tp}(\operatorname {acl}(a))$ must be always isomorphic in a G-compact T. In general, the answer turns out to be negative (see Section 3), but we obtain some positive results if, instead of looking at $\operatorname {acl}(a)$ , we add only a finite part of it to a.

For the whole Section 2, we fix the following notation.

Let us recall basic definitions and facts on covering maps between topological groups (see for example [Reference Hofmann and Morris6]).

If T is G-compact then the Lascar equivalence E is $\emptyset $ -type-definable (see [Reference Kim7, Proposition 20]), and we can assume any formula $\varphi $ in E is symmetric (i.e., $\varphi (\bar z,\bar w)\models \varphi (\bar w,\bar z)$ ).

Proof Notice that the type

$$ \begin{align*}E(xy,x'y')\wedge\exists zwz'w'\equiv xyx'y' \left(\bigvee_{ 1\leq i\ne j\leq N} (E(zw,ac_i)\wedge E(z'w',ac_j))\right)\end{align*} $$

is inconsistent, and choose $\alpha (xy,x'y')\in E(xy,x'y')$ so that the type

$$ \begin{align*}\alpha(xy,x'y')\wedge\exists zwz'w'\equiv xyx'y' \left(\bigvee_{1\leq i\ne j\leq N} (E(zw,ac_i)\wedge E(z'w',ac_j))\right)\end{align*} $$

is inconsistent. Now, if $\bar p(xy)\wedge \bar p(x'y') \wedge \alpha (xy,x'y')\wedge E(x,x')$ holds, then we can find $c'a'c''$ such that $ac'a'c'' \equiv xyx'y'$ , and then $a'\equiv ^L a$ so $a'c'' \equiv ^L ac_i$ for some i, and also $ac' \equiv ^L ac_j$ for some j, but, by the choice of $\alpha $ , $i=j$ so $ac'\equiv ^L a'c''$ , so $xy \equiv ^L x'y'$ .⊣

The following theorem and its proof will use the setting described in Notation 2.2.

Proof Assume T is G-compact. Since E is transitive, there is $\phi (xy,x'y')\in E(xy,x'y')$ such that

$$ \begin{align*}\phi(xy,zw)\wedge \phi(x'y',zw)\vdash \alpha(xy,x'y')\ \ (*),\end{align*} $$

where $\alpha \in E$ is the formula given by Lemma 2.5.

Let $(a_{\ell })_{\ell \in I}$ be a small set of representatives of E-classes of realizations of p. For any $a'c'\models \bar {p}$ , there are $\ell \in I$ and $1\leq j\leq N$ such that $\models E(a'c',a_{\ell }c_j^{a_{\ell }})$ , so $\models \phi (a'c',a_{\ell }c_j^{a_{\ell }})$ . Hence, by compactness, there are $a_0,\dots ,a_k\models p$ such that

$$ \begin{align*}\bar{p}(xy) \vdash \bigvee\{ \phi(xy,a_ic^{a_i}_j)\mid i\leq k;\ 1\leq j\leq N \}\ \ (**).\end{align*} $$

Claim 2.7. Let $f/\operatorname {Autf}^1(\bar {p})\in K$ . For each $a_ic_j^{a_i}$ chosen above, there is a unique $a_ic^{a_i}_{j'}$ with $ 1\leq j'\leq N$ such that $\phi (f(a_ic^{a_i}_j), a_ic^{a_i}_{j'})$ holds. Such a $j'$ does not depend on the choice of a representative of $f/\operatorname {Autf}^1(\bar {p})$ .

Proof Notice that $f/\operatorname {Autf}^1(\bar {p})\in K$ implies $E(f(a_i),a_i)$ , so $E(f(a_ic^{a_i}_j), a_ic^{a_i}_{j'})$ holds for some $j'$ . Now if $\phi (f(a_ic^{a_i}_j), a_ic^{a_i}_{j''})$ holds as well, then due to $(*)$ (with the symmetry of the formulas) and Lemma 2.5, we must have $j'=j''$ . The second statement of the claim follows similarly.⊣

Claim 2.8. Let $f/\operatorname {Autf}^1(\bar {p})$ , $g/\operatorname {Autf}^1(\bar {p})\in K$ . Assume that the permutations of tuples $a_ic_j^{a_i}$ by f and g described in the claim above are the same. Then $f/\operatorname {Autf}^1(\bar {p})=g/\operatorname {Autf}^1(\bar {p})$ .

Proof Let $a'c'\models \bar p$ . By $(**)$ , there is some $a_ic_j^{a_i}$ such that $\phi (a'c', a_ic_j^{a_i})$ holds. Hence, $\phi (f(a'c'), f(a_ic_j^{a_i}))$ and $\phi (g(a'c'), g(a_ic_j^{a_i}))$ hold. Moreover, by the previous claim with our assumption, there is $ j'\leq N$ such that $\phi (f(a_ic^{a_i}_j), a_ic^{a_i}_{j'})$ and $\phi (g(a_ic^{a_i}_j), a_ic^{a_i}_{j'})$ hold.

Then, again due to $(*)$ and Lemma 2.5, $f(a_ic^{a_i}_j)\equiv ^L g(a_ic^{a_i}_j)$ since $f(a_i)\equiv ^Lg(a_i)\equiv ^L a_i$ . Hence, there is $h\in \operatorname {Autf}^1(\bar p)$ such that $f(a_ic^{a_i}_j)=hg(a_ic^{a_i}_j)$ . Now by $(*)$ again, $\alpha (f(a'c'), hg(a'c'))$ holds, and, since $f(a')\equiv ^Lhg(a')\equiv ^L a'$ , we have $f(a'c')\equiv ^L hg(a'c')\equiv ^L g(a'c')$ . We conclude that $f/\operatorname {Autf}^1(\bar {p})=g/\operatorname {Autf}^1(\bar {p})$ .⊣

As there are only finitely many permutations of $a_ic_j^{a_i}$ ( $i\leq k$ , $ 1\leq j\leq N$ ), K is finite.⊣

By Remark 2.4, we have the following.

Example 2.10. Consider the following model:

$$ \begin{align*}M=((M_1,S_1,\{g^1_n\mid 0< n\}),(M_2,S_2,\{g^2_n\mid 0< n\}),\delta),\end{align*} $$

a 2-sorted structure. Here $M_1, M_2$ are disjoint unit circles. For $i=1,2$ , $S_i$ is a ternary relation on $M_i$ such that $S_i(a,b,c)$ holds iff $a,b,c$ are distinct and b comes before c going clockwise around $M_i$ from a; and $g^i_n$ is the clockwise rotation of $M_i$ by $\frac {2\pi }{n}$ -radians. The map $\delta :M_1\to M_2$ is a double covering, i.e., if we identify each $M_i$ as the unit circle in $xy$ -plane centered at $0$ , then $\delta $ is given by $(\cos t, \sin t)\mapsto (\cos 2t,\sin 2t).$ By arguments similar to those described in [Reference Dobrowolski, Kim and Lee3] for $M_i$ , it follows that $T=\operatorname {Th}(M)$ is G-compact, and, in T, $\emptyset =\operatorname {acl}^{\operatorname {eq}}(\emptyset )$ . Let ${\mathcal M}_1,{\mathcal M}_2$ be saturated models of $M_1,M_2$ respectively. For any $a_i,a^{\prime }_i\in {\mathcal M}_i$ , we have $a_i\equiv a^{\prime }_i$ ; and $a_i\equiv ^La^{\prime }_i$ iff they are infinitesimally close. Now, given $a\in M_2$ , there are two antipodal $c_1,c_2\in M_1$ such that $\delta (c_i)=a$ . Then $c_i\in \operatorname {acl}(a)$ and $ac_1\equiv ac_2$ , but $ac_1\not \equiv ^L ac_2$ . For $p=\operatorname {tp}(a)$ and $\bar p=\operatorname {tp}(ac_1)$ , $\pi :\operatorname {Gal}_{\operatorname {L}}^{1}(\bar p)\to \operatorname {Gal}_{\operatorname {L}}^{1}(p)$ is a 2-covering homomorphism of circle groups. Notice that for any $n>2$ ,

$$ \begin{align*}\delta(y)=x \wedge \delta(y')=x' \wedge (S_1(y,y',g_n^1(y)) \vee S_1(y',y,g_n^1(y')))\end{align*} $$

serves as the formula $\alpha (xy,x'y')$ in Lemma 2.5.

We have a partial answer to the second question above. Namely, we also get that K is finite when we replace the assumption of G-compactness by abelianity of $\operatorname {Gal}_{\operatorname {L}}^1(\bar {p})$ .