Interpreting groups and fields in some nonelementary classes

by Hyttinen and Lessmann and Shelah. [HLSh:821]
J Math Logic, 2005
This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem: Theorem : Let {C} be a large homogeneous model of a stable diagram D . Let p, q in S_D(A), where p is quasiminimal and q unbounded. Let P = p({C}) and Q = q({C}) . Suppose that there exists an integer n < omega such that dim(a_1 ... a_{n}/A cup C)=n, for any independent a_1, ..., a_{n} in P and finite subset C subseteq Q, but dim(a_1 ... a_n a_{n+1}/A cup C) <= n, for some independent a_1, ...,a_n,a_{n+1} in P and some finite subset C subseteq Q . Then {C} interprets a group G which acts on the geometry P' obtained from P . Furthermore, either {C} interprets a non-classical group, or n = 1,2,3 and If n = 1 then G is abelian and acts regularly on P' . If n = 2 the action of G on P' is isomorphic to the affine action of K rtimes K^* on the algebraically closed field K . If n = 3 the action of G on P' is isomorphic to the action of PGL_2(K) on the projective line {P}^1(K) of the algebraically closed field K .

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