Simple groups of finite morley rank with a tight automorphism whose centralizer is pseudofinite

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2009
Uğurlu, Pınar
This thesis is devoted to the analysis of relations between two major conjectures in the theory of groups of finite Morley rank. One of them is the Cherlin-Zil'ber Algebraicity Conjecture which states that infinite simple groups of finite Morley rank are isomorphic to simple algebraic groups over algebraically closed fields. The other conjecture is due to Hrushovski and it states that a generic automorphism of a simple group of finite Morley rank has pseudofinite group of fixed points. Hrushovski showed that the Cherlin-Zil'ber Conjecture implies his conjecture. Proving his Conjecture and reversing the implication would provide a new efficient approach to prove the Cherlin-Zil'ber Conjecture. This thesis proposes an approach to derive a proof of the Cherlin-Zil'ber Conjecture from Hrushovski's Conjecture and contains a proof of a step in that direction. Firstly, we show that John S. Wilson's classification theorem for simple pseudofinite groups can be adapted for definably simple non-abelian pseudofinite groups of finite centralizer dimension. Combining this result with recent related developments, we identify definably simple non-abelian pseudofinite groups with Chevalley or twisted Chevalley groups over pseudofinite fields. After that in the context of Hrushovski's Conjecture, in a purely algebraic set-up, we show that the pseudofinite group of fixed points of a generic automorphism is actually an extension of a Chevalley group or a twisted Chevalley group over a pseudofinite field.

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Citation Formats
P. Uğurlu, “Simple groups of finite morley rank with a tight automorphism whose centralizer is pseudofinite,” Ph.D. - Doctoral Program, Middle East Technical University, 2009.