TRANSLATION GENERALIZED QUADRANGLES FOR WHICH THE TRANSLATION DUAL ARISES FROM A FLOCK

It is shown that each finite translation generalized quadrangle (TGQ) S, which is the translation dual of the point-line dual of a flock generalized quadrangle, has a line [infinity] each point of which is a translation point. This leads to the fact that the full group of automorphisms of S acts 2-transitively on the points of [infinity], and the observation applies to the point-line duals of the Kantor flock generalized quadrangles, the Roman generalized quadrangles and the recently discovered Penttila-Williams generalized quadrangle. Moreover, by previous work of the author, the non-classical generalized quadrangles (GQʹs) which have two distinct translation points, are precisely the TGQʹs of which the translation dual is the point-line dual of a non-classical flock GQ. We emphasize that, for a long time, it has been thought that every non-classical TGQ which is the translation dual of the point-line dual of a flock GQ has only one translation point. There are important consequences for the theory of generalized ovoids (or eggs) in PG(4n - 1,q), the study of span-symmetric generalized quadrangles, derivation of flocks of the quadratic cone in PG(3,q), subtended ovoids in generalized quadrangles, and the understanding of automorphism groups of certain generalized quadrangles. Several problems on these topics will be solved completely.