On groups of finite Morley rank with a split BN-pair of rank 1

We study groups of finite Morley rank with a split BN-pair of Tits rank 1 in the case where the normal complement to B∩N in B is infinite and abelian. For such groups, we give conditions ensuring that the standard action of G on the cosets of B is isomorphic to the natural action of PSL2(F) on P1(F) for F an algebraically closed field. In particular, we show that SL2(F) and PSL2(F) are the only infinite quasisimple L⁎-groups of finite Morley rank possessing a split BN-pair of Tits rank 1 where the normal complement to B∩N in B is infinite abelian. Our approach is through the theory of Moufang sets and is tied to work attempting to classify the abelian Moufang sets of finite Morley rank.