Simple Groups of Finite Morley Rank of Odd and Degenerate Type

OF THE DISSERTATION Simple Groups of Finite Morley Rank of Odd and Degenerate Type by Jeffrey Burdges Dissertation Director: Professor Gregory Cherlin The present thesis aims to drive Borovik’s program towards its final endgame, and to lay out a plan for bringing it to a conclusion. Borovik’s program is an approach to the longstanding Algebraicity Conjecture of Cherlin and Zilber, which proposes that all simple groups of finite Morley rank are algebraic. We develop a new characteristic zero notion of unipotence and use it to make progress on various aspects of this algebraicity conjecture. Specifically, we remove the assumption of the noninvolvement of “bad fields,” known as tameness, from several points in the existing odd type theory. Tameness is removed from two critical applications within the signalizer functor theory used in the proof of Borovik’s trichotomy theorem, a result which provides the basic case division in odd type. Tameness is also removed from several points in the analysis of minimal connected simple groups. We also advance the study of groups of degenerate type which are not bad groups, by providing them with an analog of the trichotomy theorem, and we answer a question of Poizat regarding the effects of the Borovik-Poizat axioms from groups of finite Morley rank, outside of the context of groups. A strategy for the conclusion of Borovik’s project, in a suitable form, is outlined in the final chapter.