A new family of extended generalized quadrangles

For every hyperovalOofPG(2,q) (qeven), we construct an extended generalized quadrangle with point-residues isomorphic to the generalized quadrangleT*2(O) of order(q-1, q+1).These extended generalized quadrangles are flag-transitive only whenq=2 or 4. Whenq=2 we obtain a thin-lined polar space with four planes on every line. Whenq=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. *1 In memory of Giuseppe Tallini.