On the geometry of sharply 2-transitive groups

We show that the geometry associated to certain non-split sharply 2-transitive groups does not contain a proper projective plane. For a sharply 2-transitive group of finite Morley rank we improve known rank inequalities for this geometry and conclude that a sharply 2-transitive group of Morley rank 6 must be of the form $K\rtimes K^*$ for some algebraically closed field $K$.