Generalized quadrangles and projective axes of symmetry (Q846764)

The authors investigate generalized quadrangles that admit at least two projective axes of symmetry. They show that if there are three such axes incident with a common point \(x\), then \(x\) is a translation point of the quadrangle. If, in particular, \(\Gamma\) is a compact connected quadrangle with topological parameters \((p,p)\), \(p\in\mathbb N\), then \(\Gamma\) is a topological translation generalized quadrangle. The case of two opposite projective axes of symmetry is also investigated, and a characterization of the dual of the symplectic quadrangle over \(\mathbb R\) or \(\mathbb C\) among compact connected quadrangles with equal topological parameters is obtained.