Generalized Quadrangles and Projective Axes of Symmetry

We investigate generalized quadrangles Γ that admit at least two projective axes of symmetry. We show that if there are three such axes incident with a common point x, then x is a translation point of Γ. In case that Γ is moreover a compact connected quadrangle with topological parameters (p, p), p ∈ N, then Γ is a topological translation generalized quadrangle. We further investigate the case of two opposite projective axes of symmetry and obtain a characterization of the dual of the symplectic quadrangle over R or C among compact connected quadrangles with equal topological parameters.