Cherlin's "Main Conjecture" from his 1979 paper "Groups of Small Morley Rank" is the following: Every simple $\omega$-stable group is an algebraic group over an algebraically closed field. Zilber was simultaneously doing similar work (as Cherlin notes). The finite Morley rank version of the conjecture is sometimes called the Cherlin-Zilber conjecture or the Algebraicity conjecture.
There is an extensive literature for the finite Morley rank case. I am not asking about the finite rank case. I am interested in the status of the conjecture for the infinite rank case. Cherlin notes that this conjecture would imply that any simple $\omega$-stable group is of finite rank – modulo the finite rank version of the conjecture, this is essentially the content of the infinite rank version of the conjecture.
Further, Cherlin notes that one could formulate a linear version of the conjecture, in which the group acts as a subgroup of the linear transformations of some vector space.
What is the status of the infinite rank version of Cherlin's "Main Conjecture"?
Edit:
See the comments made by S. Thomas below. (Thanks for the clarification Simon +1).
Best Answer
I think the 'status' might be described as : Pillay has shown using used Selah's work that the free group is not CM-trivial.
All known counterexamples to Zilber's conjecture are CM-trivial. A non-abelian simple group of finite Morley rank is not CM-trivial. We therefore suspect that the current methods based upon Hrushovski's counterexamples cannot produce even an infinite rank counterexample who is a simple group.