The Universal Covers of a Family of Extended Generalized Quadrangles

A sufficient condition for the simple connectedness is given for an infinite family of extended generalized quadrangles Y(S) of order (q + 1,q − 1) constructed in [7] from a family S of planes inPG(5,q) with some conditions. Applying this, Y(S) is shown to be simply connected when S is obtained from a (q + 1)-arc inPG(3,q) except forq = 4, and when S is constructed from the hyperovals for which the associated permutation polynomials are explicitly given in the list [6, p.299]}, except possibly for a class of Payne.