A new trichotomy theorem for groups of finite Morley rank

The Algebraicity Conjecture for simple groups of finite Morley rank, also known as the Cherlin-Zilber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last 15 years, the main line of attack on this problem has been the so-called Borovik program of transferring methods from finite group theory. This program has led to considerable progress; however, the conjecture itself remains decidedly open. In the formulation of this program, groups of finite Morley rank are divided into four types, odd, even, mixed, and degenerate, according to the structure of their Sylow 2-subgroup. For even and mixed type the Algebraicity Conjecture has been proven, and connected degenerate type groups are now known to have trivial Sylow 2-subgroups [BBC06]. Here we concern ourselves with the ongoing program to analyze a minimal counterexample to the conjecture in odd type, where the Sylow 2-subgroup is divisible-abelian-by-finite. The present paper lies between the high Prüfer rank, or generic, case of the Borovik program, where general methods are used heavily, and the “end game” where general methods give way to consideration of special cases. In the first part, the Generic Trichotomy Theorem [Bur06] says that a minimal non-algebraic simple group of finite Morley rank has Prüfer rank at most two. Thus we may consider small groups whose simple sections are restricted to PSp4, G2, PSL3, and PSL2. In the next stage, we hope to proceed via the analysis of components in the centralizers of toral involutions; however, the existence of