A new family of extended generalized quadrangles

For every hyperoval O of PG(2.q) (q even). we construct an extended generalized quadrangle with point-residues isomorphic to the generalized quadrangle T-2(*)(O) of order (q - I, q + l). These extended generalized quadrangles are Rag-transitive only when q = 2 or 4. When q = 2 we obtain a thin-lined polar space with four planes on every line. When q = 4 we obtain one of the geometries discovered by Yoshiara [28]. That geometry is produced in [28] as a quotient of another one. which is simply connected, constructed in [28] by amalgamation of parabolics. In this paper we also give a 'topological' construction of that simply connected geometry. (C) 1997 Academic Press Limited.