Interpreting groups and fields in some nonelementary classes

Journal of Mathematical Logic 5 (1):1-47 (2005)
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This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem: Theorem. Let [Formula: see text] be a large homogeneous model of a stable diagram D. Let p, q ∈ SD, where p is quasiminimal and q unbounded. Let [Formula: see text] and [Formula: see text]. Suppose that there exists an integer n < ω such that [Formula: see text] for any independent a1, …, an ∈ P and finite subset C ⊆ Q, but [Formula: see text] for some independent a1, …, an, an+1 ∈ P and some finite subset C ⊆ Q. Then [Formula: see text] interprets a group G which acts on the geometry P′ obtained from P. Furthermore, either [Formula: see text] 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 ⋊ K* on the algebraically closed field K. •If n = 3 the action of G on P′ is isomorphic to the action of PGL2 on the projective line ℙ1 of the algebraically closed field K. We prove a similar result for excellent classes.



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References found in this work

Unidimensional theories are superstable.Ehud Hrushovski - 1990 - Annals of Pure and Applied Logic 50 (2):117-137.
Minimale Gruppen.Joachim Reineke - 1975 - Mathematical Logic Quarterly 21 (1):357-359.
Strong splitting in stable homogeneous models.Tapani Hyttinen & Saharon Shelah - 2000 - Annals of Pure and Applied Logic 103 (1-3):201-228.
Classification Theory and the Number of Nonisomorphic Models.S. Shelah - 1982 - Journal of Symbolic Logic 47 (3):694-696.

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