Niveau: Supérieur, Doctorat, Bac+8
CONSTRUCTING AN ALMOST HYPERDEFINABLE GROUP ITAY BEN-YAACOV, IVAN TOMASˇIC, AND FRANK O. WAGNER Abstract. This paper completes the proof of the group configuration theorem for simple theories started in [BY00]. We introduce the notion of an almost hy- perdefinable (poly-)structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk. The group configuration theorem is one of the cornerstones of geometric stability theory. It has many variants, stating more or less that in a stable theory, if some dependence/independence situation exists, then there is a non-trivial group behind it. In a one-based theory, any non-trivial dependence/independence situation gives rise to a group. One should consult [Pil96] for these results. The obvious question from a simplicity theorist's point of view would be how much of this can we prove in a simple theory? In the stable case, the proof could be decomposed into two main steps: 1. Obtain a generic group chunk whose elements are germs of generic functions, and whose product is the composition. 2. Apply the Weil-Hrushovski generic group chunk theorem. The second step was generalised to simple theories by the third author in [Wag01]. In [BY00], the first author tried to generalise the first, with limited success.
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