Characterization of some 4-gonal configurations of Ahrens-Szekeres type

Motivated by the Ahrens-Szekeres Quadrangles, we shall present a variation of the 4-gonal family method of construction introduced by Kantor in 1980. The relation between generalized quadrangles of order (s,s) and of order (s−1,s+1) has been known for a long time. A geometrical description of this interrelation was given by Payne in 1971 and rests on the notion of regular points or of regular lines. In this paper we wish to develop these connections algebraically in the hope of getting more insight into them from the group-theoretical point of view. In this way we are able to characterize two classes of known 4-gonal configurations.