Slanted symplectic quadrangles
The paper investigates slanted symplectic quadrangles. A slanted symplectic quadrangle over a field \(F\) is constructed from the symplectic quadrangle over \(F\) as follows: The points of the slanted quadrangle are all points of the symplectic quadrangle not collinear with a given point \(s\) of the symplectic quadrangle, the lines are all lines of the symplectic quadrangle not incident with \(s\) plus all lines of the underlying projective space that are incident with \(s\) but that are not lines of the symplectic quadrangle. Lines of the first kind are called absolute lines, lines of the second kind are called special lines. The construction does not depend on the choice of \(s\). For finite fields this construction is well-known. The slanted symplectic quadrangle over \(F\) is represented by the group \(\Gamma= (\text{PSp}_ 4 F)_ s\), i.e., the group \(\Gamma\) acts transitively on the point set and the stabilizer \(\Gamma_ p\) of a point \(p\) fixes the special line through \(p\) and acts transitively on the absolute lines through \(p\). If the field has at least five elements, then the group \(\Delta\) of automorphisms of the slanted symplectic quadrangle over \(F\) is the semidirect product of the group \(\Gamma\) with the group of automorphisms of \(F\). Moreover, the group \(\Delta\) acts transitively on the point set and has precisely two orbits in the set of lines, namely the set of special lines and the set of absolute lines. In particular, this implies that the slanted symplectic quadrangle over a field with at least five elements is not a Moufang quadrangle. For slanted symplectic quadrangles as well as for the dual quadrangles of slanted symplectic quadrangles the groups of (even) projectivities are computed. It turns out that if \(F\) is an infinite field, then the group of even projectivities of the slanted symplectic quadrangle over \(F\) contains the finitary symmetric group on \(F\), i.e., the group of all permutations moving only finitely many elements of \(F\). This implies that if \(F\) is an infinite field, then there are no non-discrete Hausdorff topologies on the point set and line set of the slanted symplectic quadrangle over \(F\) such that it becomes a topological quadrangle, i.e., a quadrangle where the geometric operations are continuous.
scientific article; zbMATH DE number 569278
scientific article