Group Configurations and Germs in Simple Theories

We develop the theory of germs of generic functions in simple theories. Starting with an algebraic quadrangle (or other similar hypotheses), we obtain an "almost" generic group chunk, where the product is defined up to a bounded number of possible values. This is the first step towards the proof of the group configuration theorem for simple theories, which is completed in [3]. Introduction. This paper represents the first step towards the proof of the group configuration theorem for simple theories, which is achieved in [3]. In its stable version, this theorem is one of the cornerstones of geometric stability theory. It has many variants, stating more or less that if some dependence/independence situation exists, then there is a non-trivial group behind it, and in a one-based theory, every non-trivial dependence/independence situation gives rise to a group (see [11]). The question of generalising it to simple theories arises naturally. In the stable case, the proof can 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 is generalised to simple theories in [13, Section 3]. This paper is concerned with the generalisation of the first step, and does so with limited success: we only obtain a generic polygroup chunk, that is a generic group chunk where product is defined only up to a bounded set of possible values. This gap is eventually filled in [3], and requires the use of altogether different tools: as far as we know, if we are not ready to go beyond hyperimaginaries and into the realm of graded almost hyperimaginaries, a generic polygroup chunk is indeed the best we can construct. In order to understand the problems arising when trying to generalise the theory of germs of generic functions to simple theories, let us first take a closer look on the stable case. There, one could define generic functions as follows: Let p be a type, q, q' be two strong types, all over the same parameters. Then p acts generically from q to q' if for some (thus any) independent realizations f ? p, x = q we have a definable f(x) ? q' such that f, x, f(x) are pairwise independent. Moreover, if p acts generically from q to q', and p' acts generically from q' to q", and p, p' are strong types, then p x p', which is the set of independent realizations of p and p', is Received June 9, 2001; revised June 26, 2002.