On small groups of finite Morley rank with a tight automorphism

We consider an infinite simple group of finite Morley rank $G$ of Pr\"{u}fer $2$-rank $1$ which admits a tight automorphism $\alpha$ whose fixed-point subgroup $C_G(\alpha)$ is pseudofinite. We prove that $C_G(\alpha)$ contains a subgroup isomorphic to the Chevalley group ${\rm PSL}_2(F)$, where $F$ is a pseudofinite field of characteristic $\neq 2$. Moreover, we prove that, if $F$ is of positive characteristic and if $-1$ is a square in $F^{\times}$, then $G \cong {\rm PSL}_2(K)$ for some algebraically closed field $K$ of characteristic $> 2$. These results are based on the work of the second author, where a new strategy to approach the Cherlin-Zilber Conjecture--stating that infinite simple groups of finite Morley rank are isomorphic algebraic groups over algebraically closed fields--was developed. In this version, we have corrected typos and clarified the proofs of some lemmas.Comment: 31 page