Definably Linear Groups of Finite Morley Rank

Introduction. Zilber’s original trichotomy conjecture proposed an explicit classification of all one-dimensional objects arising in model theory. At one point, classifying the simple groups of finite Morley rank was viewed as a subproblem whose affirmative answer would justify this conjecture. Zilber’s conjecture was eventually refuted by Hrushovski [9], and the classification of simple groups of finite Morley rank remains open today. However, these conjectures hold in two significant cases. First, Hrushovski and Zilber prove the full trichotomy conjecture holds under very strong geometric assumptions [10], and this suffices for various diophantine applications. Second, the Even & Mixed Type Theorem [1] shows that simple groups of finite Morley rank containing an infinite elementary abelian 2-subgroup are Chevellay groups over an algebraically closed field of characteristic two. In this paper, we clarify some middle ground between these two results by eliminating involutions from simple groups which are definably embedded in a linear group over an algebraically closed field in a structure of finite Morley rank, and which are not Zariski closed themselves. One may simplify terminology by saying that G is a definably linear group over a field k of finite Morley rank, implicitly using some expansion of the field language, or just a definably linear group of finite Morley rank.