A new family of extended generalized quadrangles (Q674621)

In this paper, the authors construct in an entirely geometric way an infinite family of rank 3 geometries extending the generalized quadrangles \(T^*_2 (O)\), \(O\) a hyperoval in the projective plane \(PG (2,q)\) for some prime power \(q\). The construction uses two hyperovals in \(PG (4,q)\): one in a plane \(\pi\), and another one (as set of planes) in the residue of some line belonging to \(\pi\). It is shown that for \(q= 2,4\), the geometries are essentially unique (independent of the choice of the hyperovals) and they admit a flag-transitive group. Also, all these extended generalized quadrangles satisfy the intersection property. For \(q> 4\), there are several non-isomorphic models and neither of them is flag-transitive. The authors also give some information on (universal) covers; in particular they determine the universal covers in the case \(q=2,4\). Some open problems are mentioned at the end of the paper.