数理解析研究所講究録
第 1555 巻 2007 年 61-69

61

THE GROUP CONFIGURATION THEOREM AND ITSAPPLICATIONSBYUNGHAN KIM

The group configuration theorem for stable theories [6] plays important roles in solving deep problems in geometric stability. The
theorem roughly says that one can get the canonical non-trivial typedefinable homogeneous space (i.e. a group with its transitive action ona set, all type-definable) from a group configuration, a certain geometrical configuration, in stable theories. Recently fruitful achievementsof the generalization of the theorem into the context of simple theories
were made. In their topical paper [1], I. Ben-Yaacov, E. Tomasic and F.
O. Wagner generalize the group configuration theorem by obtaining aninvariant group from the group configuration in simple theories. However the group they produce does not completely fit into the first-ordercontext. On the other hand, T. de Piro, B. Kim and J. Millar succeed
in getting the canonical hyperdefinable group from the group config$th\infty ry$

uration under 4-amalgamation in simple theories [5]. The element of
the group is a hyperimaginary, an equivalence class of a type-deflnableequivalence relation, and the group operation is typedefinable, hence
the group belongs to the domain of the standard first-order logic. The
former result is for all simple theories but the group obtained is non
hyperdefinable, where as the latter producing the desirable hyperdefinable group has a pay-off of an assumption of generalized amalgamation.
In this small note, we will review the latter result of de Piro, Kim andMillar, together with the notions around generalized amalgamation.
(There is a nice expository paper on the former result appeared in the
Bulletin of Symbolic Logic [2].) Kim recently continue the construction
and complete the group configuration theorem [13]. Namely, under 4amalgamation, he is able to construct a hyperdefinable homogeneous
space equivalent to the given group configuration. This will be reviewed
too. Next, we will speak about its applications. In particular we mainly
pay our attentions to the open problem whether pseudolinearity implies
linearity, which is known to be true for stale theories.
We assume that the reader is familiar with basics of simplicity theory [19]. Throughout the paper, $T$ is a complete simple theory. We
Byunghan Kim was supported by a KRF grant 2006-312-C00455.

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BYUNGHAN KIM

of with hyperimaginaries, andwork in a saturated modelare (possibly infinitary) hyperimaginaries, $M,$ $N$ are small elementary
are also hyperimaginaries). As
submodels. (Note that tuples fromhave the same type (Lascar strong
usual, $a\equiv Ab(a\equiv LAb)$ means
type, resp.) over . We point out that usually $bdd(a)$ denotes theset of all countable hyperimaginaries definable over $a[19,3.1.7]$ . Here,depending on the context, it can be either a specific sequence which
linearly orders the set $bdd(a)$ ; or, since a sequence of hyperimaginaries
is again a hyperimaginary (of a large arity), a fixed hyperimaginaryinterdefinable with the sequence.
$T$$\mathcal{M}$$a,$ $b,$$\ldots$$\mathcal{M}^{eq}$

$a,$

$b$

$A$

1. GENERALIZED TYPE-AMALGAMATIONAs well-known, in [14], B. Kim and A. Pillay prove the following formof type-amalgamation (or the independence theorem) for all simple
theoriae.
, andType-Amalgamation 1.1. If, then there is such thatis B-independent.
and:$a_{1}^{\lambda_{B}}a_{2},$$d\equiv^{L}d_{i}Ba$$d$$d_{2}$$d_{i}^{\lambda_{B}}a_{i}(i=1,2)$$d_{1}\equiv^{L}B$$\{d, a_{1}, a_{2}\}$

Before Kim and Pillay’s work, the original type-amalgamation isstated and proved to be held in some simple algebraic structures in acouple of papers by Hrushovski [8] [10]. In particular, the one stated in[8] (which is written earlier than [14] but published later) is as follows.
Type-Amalgamation 1.2. Suppose that there are complete types$r_{i}(x_{i})(i=1,2,3)$ and $r_{jk}(x_{jk})(1\leq j<k\leq 3)$ , all over a set, wherevariables,such thatis possibly an infinite set of$B$$x_{i}$(1)$x_{j}\cup x_{k}\subset x_{jk}$and$r_{j}\cup r_{k}\subset r_{jk}$

, and

says
(2)
are B-independent,
and(3)
is as a set $bdd(x_{j}x_{k}B)$ .
Then there is a complete typethatis B-independent, and$r_{jk}(x_{jk})$$x_{k}$$x_{j}$$x_{jk}$$r_{123}(x_{123})\supseteq r_{12}\cup r_{13}\cup r_{23}$$\{x_{1}, x_{2},x_{3}\}$$x_{123}$

is

over

$bdd(x_{1}x_{2}x_{3}B)$$B$.

saying

It is not difficult to see that the two statements 1.1 and 1.2 are equivalent. However this equivalence no longer holds if tbe index increaees.

Generalized Type-Amalgamation 1.3. For B-independent $A=$andwhere Afor $i=1,$ $..,n-1$ ,$A_{ij}=A_{i}\cap
A_{j}$, then there is such thatwheneverwhereis B-independent.
and$\{a_{1}, \ldots, a_{n-1}\}$$d_{i^{\backslash }}L_{B}A_{i}$$d_{i}\equiv^{L}BA:fd_{j}$$d\equiv LBA_{i}d_{i}$$=A\backslash \{a_{i}\}$$d$$\{d, a_{1}, \ldots, a_{n-1}\}$

Generalized Type-Amalgamation 1.4. Let $W$ be a collection ofsubsets of $\{1, \ldots,n\}=u_{n}$ , closed under subsets. For each $w\in W$ ,

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THE GROUP CONFIGURATION THEOREM AND ITS APPLICATIONS

complete typeover $B$ is given whereis possibly an infinite
set of variables. Suppose that(1) for.
and,Moreover for any(2) $\{a_{\{i\}}|i\in w\}$ is B-independent,(3)(and the map $a_{w}arrow x_{w}$ is a
is as a set
bijection).
Then there is a complete typeover $B$ such that (1),(2),(3)$r_{w}(x_{w})$$w\subseteq w’,$$x_{w}$$x_{w}\subseteq x_{w’}$$r_{w}\subseteq r_{w’}$$a_{w}\models r_{w}$$bdd(\bigcup_{i\in w}a_{\{i\}}B)$$a_{w}$$r_{u_{n}}(x_{u_{*}},)$

hold for all $w\in W\cup\{u_{n}\}$ .

An example shows that the two propositions do not coincide evenwhen $n=4$ .

, choose distinct
Example 1.5. In the random graph $M$ in$a_{i},b_{i},c_{i}\in M$ and imaginary elements $d_{i}=\{b_{i}, q\}(i=1,2,3)$ .
We,
additionally assume that$tp(a_{1}a_{2};b_{3}c_{3})=tp(a_{2}a_{3};b_{1}c_{1})=tp(a_{1}a_{3};b_{2}c_{2})$and. Now it follows
that for $\{i,j, k\}=\{1,2,3\},$. But it iv easy to see that
$Lstp(d_{2}/a_{1}a_{3})$ , and Lstp
Lstp
have no common re$M$
alization. Namely,
does not satisfy 1.3 for $n=4$ (and larger).
However, due to elimination of weak imaginaries (and eliminationof hyperimagnaries) of the random graph, for any hyperimaginary $B$ ,$bdd(B)=dc1(A)$ for a set $A$ in a home sort. Hence to check 1.4, it
suffices to examine the amalgamation in the home-sort. It follows that
$M$ satisfies 1.4 for every. The reader may wonder why the above
arrangement ofdoes not raise a trouble as before. If we put$r_{\{i\}}=tp(a_{i})(i=1,2,3)$ , and $r_{\{4\}}=tp(b_{i}c_{i})$ , then we should let $r_{\{1,4\}}=$$tp(a_{1}; b_{3}c_{3})=tp(a_{1}; b_{2}c_{2}),$ $r_{\{3,4\}}=tp(a_{3};b_{2}c_{2})=tp(a_{3};b_{1}c_{1})$ (notethat $ac1(d_{i})=dc1(b_{i}c_{i}))$ . But thenmust be either $tp(a_{2};b_{1}c_{1})$$tp(a_{2};b_{3}c_{3})$, which are distinct!, i.e. the arrangement does not giveora compatible system of types.
$\mathcal{L}=\{R\}$$R(a_{1}, c_{3})\wedge R(a_{2}, b_{3})\wedge\urcorner R(a_{1}, b_{3})\wedge\neg R(a_{2},c_{3})$$d_{i}\equiv_{a_{j}}^{L}d_{k}$$(d_{1}/a_{2}a_{3}),$$(d_{3}/a_{1}a_{2})$$n$$a_{i},$$b_{i},q$$r_{\{2,4\}}$$r_{w}(x_{w})$The example also says 1.3 is not preserved in the interpreted theories
while 1.4 is. It is generally agreed that 1.4 is the correct definition ofgeneralized amalgamation.

Definition 1.6. We say has n-complete amalgamation (n-CA) if 1.4
holds for . We simply say has n-amalgamation if 1.4 holds for
$T$

$n$

with

$T$

$W=\mathcal{P}(u_{n})^{-}=\mathcal{P}(u_{n})\backslash \{u_{n}\}$$n$.

Clearly k-CA implies n-CA for $k\geq n$ . Note that 4-CA and 4amalgamation are equivalent. For each $n\geq 3$ , there are examples
having n-CA but not having $(n+1)- CA[16]$ . Stable theories satisfy
n-CA over models, but not in general over algebraically closed sets [5].

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BYUNGHAN KIM

This unsatisfactory phenomenon leads to define the so-called model-n$CA$ , a variation of n-CA, which all stable theories have. We omit thedescription, but for the detail, see [5] or [15].

In this note, as we will concentrate our attentions to 4-amalgamation,
we restate it in the similar manner of 1.3 which seems helpful to conceptualize.

4-Amalgamation 1.7. Let $\{i,j, k\}=\{1,2,3\}$ . Suppose that, all boundsuch thatandindependent
be some enumerations of $bdd(a_{i}d_{j})$ ,edly closed, are given. Let$bdd(a_{i}a_{j})$ , respectively.
, then there isIf, such that for $i<j$ ,, and enumerations
with$a_{0^{-}}$$d_{i}\downarrow_{a_{0}}a_{j}a_{k}$$\{a_{1}, a_{2}, a_{S}\}$$a_{0}\subseteq a_{i},$$d_{i}$$\overline{a_{i}\underline{d_{j}}},\overline{a_{i}a_{j}}$$d_{j}a_{i}\equiv_{a;}\overline{d_{k}a_{i}}$$d(\lambda_{a0}a_{1}a_{2}a_{3})$$\overline{da_{i}}$$d\equiv_{ao}d_{i}$$\overline{da_{i}}\overline{da_{j}}\equiv\overline{d_{k}a_{i}}\overline{d_{k}a_{j}}\overline{h^{a}j}$

Before closing this section, we point out the notion of n-simplicity,
initially introduced by A. Kolesnikov [16] and further modifications
were made in [15]. Recall that, 1.1 is proved by the use of the following
fact.
is a Morleysimple.) Assume that
Fact 1.8.
dbthen there is $d\equiv ba0c$ such thatsequence over . If.
indiscemible and$I=\langle a_{n}|n\in\omega\rangle$$\int T$$b$

$I$$c^{L_{b}}\backslash a_{0_{f}}$$i^{q}$$d\backslash L_{b}I$The property 1.8 proved in [12] is indeed a special case of type$amalg_{\bm{t}1}ation$ ( $=3$-amalgamation). In other words, the particular amalgamation property implies full 3-amalgamation. Thus it natural to
ask whether a higher dimensional variation of 1.8 can imply generalized amalgamation. Indeed Kolesnikov proved in [16] that the following
proper.ty, a particular case of 1.3 for $n=4$ , implies it.
$is$is a Morley sequence
Property 1.9. Assume that $I=$ \langle I$d\equiv ba0a_{1}c$ such thatandover . If’ then there is.
is db-indiscernible and$a_{n}$$b$$c^{L_{b}}\backslash a_{0}a_{1}$$n\in\omega\rangle$$I$$a_{0}\equiv^{L}a_{1}bc$$d\backslash L_{b}I$

But as 1.4 is the correct notion of amalgamation, not 1.3, 1.9 has tobe modified appropriately indicating a special case of 4-amalgamation.
The modified property, whicb we call 2-simplicity, is equivalent to 4amalgamation as Kolesnikov’s idea goes through in this contut [15].
But the question remains whether it keeps holding for larger . Surprisingly, it is not unless n-simplicity for $n\geq 3$ should be defined interms of finite Morley sequences rather than infinite ones. For details,see [5] or [15].
$n$65THE GROUP CONFIGURATION THEOREM AND ITS APPLICATIONS

2. THE GROUP CONFIGURATION THEOREM
Definition 2.1. By a group configumtion we mean a 6-tuple of hyperimaginaries $C=(f_{1}, f_{2}, f_{3}, x_{1}, x_{2}, x_{3})$ over a hyperimaginary suchthat, for $\{i,j, k\}=\{1,2,3\}$ ,(1) $f_{i}\in bdd(f_{j}, f_{k};e)$ ,(2) $x_{i}\in bdd(f_{j}, x_{k};e)$ ,(3) all other triples and all pairs from $C$ are independent over .
$e$

$e$

$f_{2}x_{1}\#^{x_{2}}x_{3}f_{1}f_{3}$If it has the property that $bdd(f_{i};e)=bdd(Cb(x_{j}x_{k}/f_{i}e);e)$ , we caJlsuch $C$ a bounded quadmngle. In particular, we callwhere $f_{i}’=Cb(x_{j}x_{k}/ef_{i})$ , an induced bounded quadrangle fram $C$ over
. Now if additionally $f_{i}\in bdd(x_{j}, x_{k};e)$ , we call the the groupconfiguration $C$ over principal. We say two group configurations$C=(f_{1}, f_{2}, f_{3}, x_{1}, x_{2}, x_{3})$ overover
andare equivalent (over d) if for someand each$Cd,$$(x_{i},x_{i}’)(i=1,2,3)$
pair ofis interbounded over . Its transitive
closure is an equivalence relation among the group configurations.
$(f_{1}’, f_{2}’, f_{3}’,x_{1},x_{2},x_{3})$

$e$

$e$

$e$

$e’$

$C’=(f_{1}’, f_{2}’, f_{3}’, x_{1}’,x_{2}’, x_{3}’)$$d\supseteq ee’,$$C’\backslash$$d$$(f_{i}, f_{i}’),$The reason why the configuration is said to be a group configurationis that it is canonically obtained from a given hyperdefinable homo$gen\infty us$ space.
More precisely, let $((G, 0),$ $X,$ ) be a hyperdefinablehomogeneous space (i.e. the hyperdefinable action, of the group on
the set $X$ is transitive) over . We say $a\in X$ is generic (over ), if$g.$for $g\in G$ withholds. For notational convenience, we
suppress to . Similarly to the group case, if $x(\in X)$ is independent
with generic $f\in G$ , then $f.x$ is generic. Hence a generic element of $X$
exists. Moreover generic $f(\in G)$ is generic with respect to $X$ as well.
Namely, for $y(\in x)\lambda_{f,y\backslash }Lf.y$ holds. We have the following.
$.$

$G$
$e$

$e$

$g\lambda_{e}a,$$e$$a$ $\lambda_{e}g$$\emptyset$

Observation 2.2. A hyperdefinable homogeneous space $(G, X)$ (over$f_{3}\in G$ and $x_{1}\in X$ , all geneis given. We can choose, suchthatis independent. Thenfoms agroup configuration where $f_{1}=f_{2}o(f_{3})^{-1},$ $x_{2}=f_{3}.x_{1},$. Notethat $x_{i}(i=1,2,3)$ are all genenc. We callgroup configuration
obtained from the homogeneous space $(G, X)$ .
$r\dot{v}c$$f_{2},$$\emptyset)$$\{f_{2}, f_{3}, x_{1}\}$$C=(f_{1}, f_{2}, f_{3}, x_{1}, x_{2}, x_{3})$$x_{3}=f_{2}.x_{1}$$f_{i},$$C_{f}$$a$

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BYUNGHAN KIM

The group configuration theorem is a theorem about the reverse
process. The theorem says that a given group configuration, one canconstruct a homogeneous space $(G, X)$ having an equivalent group configuration. In [5], de Piro, Kim and Millar obtained the first step of
the theorem. Namely, given a group configuration, a canonical groupwhose generic elements are equivalent to the first triple of the configuration. Then recently Kim [13] completes the group configuration
theorem under 4-amalgamation.
Theorem 2.3. (The group conflguration theorem) Assume
has 4-CA. Aft er possibly naming a model, we can assume.
$C$an induced bounded quadmngle from a group configuration
over we can construct a hyperdefinable homogeneous space oversuch that $C$ and a bounded quadrangle obtained fivm the space are
equivalent.
$T$$\emptyset=bdd(\emptyset)$$Give\grave{n}$$\emptyset$$\emptyset_{f}$Recall that for any stable , if the elements of the group configuration is finitary, then a type-definable homogeneous space having anequivalent configuration is conItructible.
$T$

3. APPLICATIONS AND PSEUDOLINEARITY

One application of 2.3 (or just the earlier version of dePiro, Kim, andMillar) is the following result. This extends the theorem [4, 3.23] that,in any modular non-trivial -categorical simple , an infinite vector$T$$\omega$space over some finite field is definably recovered in. Recall that$T$
is said to be $non- t\dot{n}vial$ if there are hyperimaginaries
andsuch that for $1\leq i<j\leq 3,$
independentareover whereas
is dependent over .
Theorem 3.1. Suppose that $T$ is modular,
haning 4-CA.
Then there is a hyperdefinable infinite bounded-by-Abelian group $V$ overa model $M$ of SU-mnk 1 generic types. Moreover for the bounded$\mathcal{M}^{eq}$$a_{1},$$A$$a_{i},$$a_{2},a_{3}$$A$$a_{j}$$A$$\{a_{1}, a_{2}, a_{3}\}$$non- t_{\Gamma\dot{b}}\tau\dot{n}al,$subgroup $V_{0}=V\cap bdd(M),$forms a vector space over the division
nng of $bdd(M)$ -endomorphisms of such that for$V/V_{0}$

$R$

$V$

$b\in bdd(a_{1}\ldots a_{n})$$\alpha_{i}\in R$.

iff$b,$ $a_{1},$$\ldots,$$b+V_{0}=\alpha_{1}(a_{1}+V_{0})+\ldots+\alpha_{n}(a_{n}+V_{0})$$a_{n}\in V_{f}$for some

Theorem 3.1 is important since it shows that from a pure logical condition of independence, we can recover a concrete algebraic structure.
The theory being modu lar simply means that the model theoretic
dimension property is similar to that of linear (projective or, affine)spaces. Namely, we say modular if
holds for any bound$B$$A,$
edly closed sets
. For finite dimensional case, it simply means
$T$

$T$

$A\backslash \llcorner_{A\cap B}B$$\dim(A\cup B)=\dim(A)+\dim(B)-\dim(A\cap B)$.

67
THE GROUP CONFIGURATION THEOREM AND ITS APPLICATIONSThis is exactly the case when a space is projective or affine. Hence thegroup configuration theorem in stable and simple theories is the main
to recover an underlying concrete algebraic structure from a structure
having a pure model theoretic condition. This issue is also related tothe so-called Zilber’s principle. We will get back to this later.

For the rest of this section, we pay our attentions to the possible
application of 2.3 to the pseudolinearity conjecture. Before stating
what it is, let us recall some of necessary definitions. We also restrict
our attentions to a solution set $D$ of SU-rank 1 Lascar strong type over,for convenience, . In a stable theory, $D$ can be (strongly) minimal.
$\emptyset$

Definition 3.2. Let $k\geq 1$ .
(1) We say $D$ is k-linear (or pseudolinear) if for any two singletons$b\in D$ and parameters $B$ with $SU(ab/B)=1,$ $SU(e)\leq k$where $e=Cb(ab/B)$ . We say $D$ is linear if it is l-linear.
(2) We say $D$ is k-based if for any indiscernible sequence.
from $D,$
is Morley over$a,$

$I=(\overline{c}_{i}|i\in\omega\rangle$$I_{k}$$I\backslash I_{k}$$:=\{\overline{c}_{i}|i<k\}$, the rank of theHence $D$ being k-linear means that any curwe inspace of its conjugates is bounded by . It is well-known, in general,
an infinite (rank 1, e.g. algebraically closed) field is not pseudolinear.
For example, if we take a curve defined by$y=x^{n}+a_{n-1}x^{n-1}+\ldots+a_{1}x+a_{0}$ ,$D^{2}$

$k$

where

$\{a_{0}, \ldots, a_{n-1}\}$is algebraically independent, then the rank of the, thus can be arbi-

space of the conjugates of the curve is possibly$\geq n$trarily large.
Now, by similar ideas in the proof of [4, 3.6], the following can beobtained.

Theorem 3.3.
(1) The following are equivalent.
(a) $D$ is k-linear.
(b) $D$ is k-based.

(2) The following are equivalent.
(a) $D$ is linear.
(b) $D$ is modular, $i.e$ . for any $A,$ $B\subseteq D,$ A $\lambda_{bdd(A)\cap bdd(B)}B$ .

As mentioned after 3.1, typical examples of $lInear(=modular)$ structures are vector spaces. Conversely, 3.1 says
being modular is thesame amount of saying that it can be reduced to an underlying vector
space. After having seen examples so far, one may boldly guess that
there is no ‘real’ k-linear examples other than $k=1$ . Namely we have$D$the following conjecture.

68
BYUNGHAN KIM

Pseudolinearity Conjecture 3.4. If $D$ is k-linear, then it is linear.

Indeed Zilber’s principle (or dichotomy) goes a little further. He
conjectured that any non-trivial strongly minimal structure is either
modular (so interpreting an infinite vector space), or interpreting afield (which has to be algebraically closed). As known, his conjecture
was shown to be false by Hrushovki who constructed counterexamples[7]. His construction method itself created an important new area inmodel theory. After then, he and Zilber together suggested a famous
Zariski condition, and under the constraint on the strongly minimal
structures, they succeeded to show the dichotomy [11]. It turns outthat this dicbotomy plays a great role in the applications of model
theory to other branches of mathematics such as geometry and number$th\bm{m}ry-[9]$ . Extending the dichotomy to the context of general simple
rank 1 set $D$ is a big open project. Theorem 3.1 can be considered
as an achievement in this direction, as it says at least for concerning
modularity, it is to do with a concrete vector space as in stable case.
Now by the remark after 3.2, if Zilber’s principle holds, then 3.4
easily follows: Nonlinearity of $D$ implies the interpretability of a field
which can not be k-linear.
But, regardless of that Zilber’s principle is false, 3.4 is known to betrue for stable theories [3]. The proof uses the group configuration
theorem for stable theories. Let us briefly review the proof. If stable $D$
is k-linear (for minimal such ), then easily a group configuration $C=$
can be obtained where $rk(f_{i})=k$ and $rk(x_{i})=1$ .
Then by the group configuration theorem, there is a type-definablehomogeneous space $(G, X)$ whose group configuration is equivalent to$C$ .
In particular, ranks are preserved. Namely $rk(G)=k$ and rk(X)
1. Then by the general stable group theory [17, 1.6.25], $rk(G)=1,2$ or
3. If 2 or 3, then an infinite field is interpretable from $X$ , which again
is not k-linear (the remark after 3.2). Hence must be 1, and 3.4 forstable theories is obtained.
When we try to mimic the ideas under 4-CA, the initial part ofthe proof will go through using 2.3, so from that $D$ being k-linear wehave a hyperdefinable homogeneous space $(G,X)$ with $rk(G)=k$ andrk(X) $=1$ . But we do not have so far the analogous theorem to [17,1.6.25]. In other words, problem is rather reduced to the theory of
hyperdefinable groups having simple theories. So far no progress wasmade in this regards. But we believe that, under 4-amalgamation, one
may develop finer group theory so that many important open problems
including this and supersimple field conjecture (any supersimple fieldis pseudo algebraically closed) can be resolved.
$k$$(f_{1}, f_{2}, f_{3}, x_{1}, x_{2}, x_{3})$

$=$

$k$

69
THE GROUP CONFIGURATION THEOREM AND ITS APPLICATIONS

We finally point out that 3.4 is proved to be true for any -categoricalsimple theories [18]. For an -categorical structure, the group constructed in [1] is definable, and for this particular case, a finer grouptheory exists.
$\omega$$\omega$

REFBRENCES[1] I. Ben-Yaacov, I. Tomasic, and F. Wagner, ‘Constructing an almost hyperdefinable group’, Joumal of Math. Logic 4 (2004) 181-212.
[2] I. Ben-Yaacov, I. Tomasic, and F. Wagner, ‘The group configuration in simple
theories and its applications’, Bulletin of Symbolic Logic 8 (2002) 28&298.

[3] S. Buechler ‘Pseudoprojective strongly minimal sets are locally projective’,$J\dot{o}umal$of Symbolic Logic 56 (1991) 1184-1194.
[4] T. de Piro, B.Kim, ‘The geometry of l-based minimal types’, }$unsactions$ ofAmerican Math. Soc., 355 (2003) 4241-4263.
[5] T. de Piro, B.Kim, and J. Millar ‘Constructing the hyperdefinable group fromthe group configuration’, will appear in Joumal of Math. Logic.
[6] E. Hrushovski, ‘Unimodular minimal theories’, Joumal of London Math. Sec.
46 (1992) 385-396.
[7] E. Hrushovski, ‘A new strongly minimal set’, Ann. of Pure and Applied Logic
62 (1993) 147-166.
$I$[8] E. Hrushovski,-finite fields and related structures’, Quademi $di$

$Pseud\triangleright$

Matematica 11 (2002) 151-212.
[9] E. Hrushovski, ‘The Mordell-Lang conjecture for function fields’, J. of AMS 9(1996) 667-690.
[10] E. Hrushovski and A. Pillay, ‘Groups definable in local fields and pseudo-finite
fields’, Israel Joumal of Math. 85 (1994) 203-262.
[11] E. Hrushovski and B. Zilber, ‘Zariski geometries’, Joumal of AMS 9 (1996)
1-56.
[12] B. Kim,‘Forking in simple unstable theories’, J. London Math. Soc. 57 (1998)

257-267.

[13] B. Kim, ‘Recovering the hyperdefinable group action in the group conflguration
theorem’, submitted.
[14] B. Kim and A. Pillay, ‘Simple theories’, Ann. Pure and Applied Logic 88 (1997)

149-164.

[15] B. Kim, A. Kolesnikov and A. Tsuboi, ‘Generalized amalgamation and nsimplicity’, will appear in Ann. Pure and Appli $ed$ Logic.
[16] A. Kolesnikov, ‘n-simple theories’, Ann. Pure and Applied Logic 131 (2005)

227-261.

[17] A. Pillay Geometric stability theory, Oxford University Press, Oxford (1996).
[18] I. Tomasic and F. O. Wagner, ‘Applications of the group configuration theorem
in simple theories’, Joumal of Math. Logic 3 (2003) 239-256.
[19] F. O. Wagner, Simple theories, Kluwer Academic Publishers, Dordrecht (2000).

DEPARTMENT OF MATHEMATICS, YONSEI UNIVERSITY, 134 SHINCHON-DONG,SEODAEMUN-GU, SEOUL 120-749, SOUTH KOREA
E-mail address: bkimQyonsei. ac. kr