Flag-transitive Extensions of Dual Affine Planes

(with 1 , q , ̀ ) . An infinite family of simply connected examples for this diagram with q 5 2 is obtained by truncating Coxeter complexes of type D n . Two dif ferent finite families with three members and q 5 4 arise from the Steiner systems for the Mathieu groups M 2 2 , M 2 3 and M 2 4 . The geometries of these two families are simply connected . Three more simply connected examples of rank 3 are described in [1] , with q 5 2 , 4 and 16 respectively . We shall prove that no more simply connected examples exist besides those mentioned above .