The Additive Collapse

In 1988 Ehud Hrushovski [Hr1] refuted Zil’bers Conjecture. He constructed strongly minimal theories with non-locally-modular geometries and without any infinite group interpretable in these theories. In [Bau2] uncountably categorical groups are constructed that have non-locally-modular geometries and do not allow the interpretation of infinite fields. The essential Amalgamation Lemma in that paper has a very long “bilinear-combinatorial” proof. The aim of this paper is to find general properties of an ω-stable theory T such that the additive collapse can be done. In [JG] Bruno Poizat gives a proof of Hrushovski’s result above. First he constructs a theory of Morley rank ω already with that kind of geometry he finally wants. In a second step he does the collapse to obtain the desired theory.