Involutions in Groups of Finite Morley Rank of Degenerate Type

Modern model theory can be viewed as a subject obsessed with notions of dimension, with the key examples furnished by linear dimension on the one hand, and the dimension of an algebraic variety (or, from another point of view, transcendences degree) on the other. There are several rigorous, and not always equivalent, notions of abstract dimension in use. For historical reasons the one we use is generally referred to as Morley rank. In the applications of model theory, it is important that this dimension may be ordinal valued, but the case of finite dimension continues to stand out. For example, in the model theoretic approach to the Manin kernel in an abelian variety, one enriches the underlying algebraically closed field with a differential field structure, at which point the abelian variety becomes infinite dimensional, but the Manin kernel itself is finite dimensional, which accounts for a certain number of its fundamental properties. For some time it was hoped that one would be able to classify the “one-dimensional” objects arising in model theory explicitly and in complete generality, a hope which was dashed by a construction of Hrushovski. But in diophantine applications, even after leaving the algebraic category, one has in addition to the dimension notion a topology reminiscent of the Zariski topology, and some very very strong axioms, given in [HZ96]. In this case one gets the desired algebraicity result, in nondegenerate cases, with substantial diophantine applications (cf. [Bo98, Sc01]).