# 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 andIf 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.

- 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}, doi = {10.1142/S0219061305000390}, mrclass = {03C45 (03C52 22F50)}, mrnumber = {2151582}, mrreviewer = {O. V. Belegradek}, doi = {10.1142/S0219061305000390}, note = {\href{https://arxiv.org/abs/math/0406481}{arXiv: math/0406481}}, arxiv_number = {math/0406481} }