Flag-transitive Extensions of Dual Affine Planes

In this paper I finish the work carried on in [10] and [1], obtaining a classification of flag-transitive geometries belonging to the following diagram: <q<∞).An infinite family of simply connected examples for this diagram withq=2is obtained by truncating Coxeter complexes of typeDn.Two different finite families with three members andq=4arise from the Steiner systems for the Mathieu groupsM22, M23andM24.The geometries of these two families are simply connected. Three more simply connected examples of rank 3 are described in [1], withq=2, 4and 16 respectively. ll prove that no more simply connected examples exist besides those mentioned above.