Finitely axiomatizable strongly minimal groups

We show that if G is a strongly minimal finitely axiomatizable group, the division ring of quasi-endomorphisms of G must be an infinite finitely presented ring. Questions about finite axiomatizability of first order theories are nearly as old as model theory itself and seem at first glance to have a fairly syntactical flavor. But it was in order to show that totally categorical theories cannot be finitely axiomatized that, in the early eighties, Boris Zilber started developing what is now known as “Geometric stability theory”. Indeed, as is often the case, in order to answer such a question, one needs to develop a fine analysis of the structure of models in the class involved and to understand exactly how each model is constructed. The easiest way to force a structure to be infinite by one first order sentence is to impose an ordering without end points, or a dense ordering, thus making the structure unstable. It was hence rather natural to wonder about theories at the other extremity of the stability spectrum, and, in the early 60’s, to ask whether there existed finitely axiomatizable totally categorical theories or simply uncountably categorical theories [22], [17]. Each model of a totally categorical theory is prime over a strongly minimal set. It is not too difficult to see that a totally categorical strongly minimal set cannot be finitely axiomatizable ([15]). Much more complicated, the proof of the non finite axiomatizability for the whole class goes through a characterization of the geometries associated to totally categorical strongly minimal sets (locally modular and locally finite) and then an analysis of how any model is “built” around the strongly minimal set ([23], [24] and [6] where the result is proved for all ω-stable ω-categorical theories). Around the same time as Zilber’s negative answer for the totally categorical case, Peretjat’kin produced an example of a finitely axiomatized א1-categorical theory [19]. This example was in the following years simplified by Baisalov (see [9, 12.2, Example 5]). This final example has Morley Rank equal to 2, thus still leaving open the question of the existence of a finitely axiomatizable strongly minimal set (Morley rank and degree equal to 1). Furthermore all the known examples of finitely axiomatizable א1-categorical theories are rather similar and constructed around a strongly minimal set with trivial