Around unipotence in groups of finite Morley rank

Abstract

The central question about groups of finite Morley rank is the Cherlin-Zil’ber Conjecture, which states that the infinite simple ones are algebraic. The search for satisfactory analogues in the context of groups of finite Morley rank of key notions in algebraic group theory has therefore been a continuing concern. Semisimple and unipotent elements are fundamental for understanding affine algebraic groups. While Carter subgroups of groups of finite Morley rank (Definition 6.5) offer a reasonably well-behaved analogue of maximal tori in algebraic groups (see [12], [14] and [21]), there has been more than one proposition for an analogue of unipotent subgroups of algebraic groups. The most recent and most effective of these has been introduced by Burdges in [7] (see section 2). In this article, following up on Burdges’ ideas, we introduce and study U -groups (Definition 5.1) as a new analogue of unipotent subgroups for groups of finite Morley rank. Our main result is: