Sh:821

• Hyttinen, T., Lessmann, O., & Shelah, S. (2005). Interpreting groups and fields in some nonelementary classes. J. Math. Log., 5(1), 1–47. arXiv: math/0406481 DOI: 10.1142/S0219061305000390 MR: 2151582
• Abstract:
  This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem:

  Theorem: Let \mathfrak{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(\mathfrak{C}) and Q = q(\mathfrak{C}). Suppose that there exists an integer n < \omega such that dim(a_1\dots a_{n}/A \cup C)=n, for any independent a_1,\dots, a_{n} \in P and finite subset C \subseteq Q, but dim(a_1\dots a_n a_{n+1}/A\cup C)\leq n, for some independent a_1,\dots,a_n,a_{n+1}\in P and some finite subset C\subseteq Q. Then \mathfrak{C} interprets a group G which acts on the geometry P' obtained from P. Furthermore, either \mathfrak{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 \mathbb{P}^1(K) of the algebraically closed field K.

• Version 2003-09-04_11 (40p) published version (47p)

@article{Sh:821,
 author = {Hyttinen, Tapani and Lessmann, Olivier and Shelah, Saharon},
 title = {{Interpreting groups and fields in some nonelementary classes}},
 journal = {J. Math. Log.},
 fjournal = {Journal of Mathematical Logic},
 volume = {5},
 number = {1},
 year = {2005},
 pages = {1--47},
 issn = {0219-0613},
 mrnumber = {2151582},
 mrclass = {03C45 (03C52 22F50)},
 doi = {10.1142/S0219061305000390},
 note = {\href{https://arxiv.org/abs/math/0406481}{arXiv: math/0406481}},
 arxiv_number = {math/0406481}
}