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: [GRAPHICS] (with 1 < q < infinity). An infinite family of simply connected examples for this diagram with q = 2 is obtained by truncating Coxeter complexes of type D-n. Two different finite families with three members and q = 4 arise from the Steiner systems for the Mathieu groups M(22), M(23) and M(24). The geometries of these two families are simply connected. Three more simply connected examples of rank 3 are described in [1], with 4 = 2, 4 and 16 respectively. We shall prove that no more simply connected examples exist besides those mentioned above. (C) 1996 Academic Press Limited