Triangular extended generalized quadrangles
A point-line incidence geometry \({\mathcal S}\) such that for every point P the incidence structure \({\mathcal S}_ P\) consisting of points joined to P and blocks containing P is a generalized quadrangle, is called an extended generalized quadrangle. It is moreover called triangular if for every point P and every line L not incident with P, P is joined to no points or exactly two points of L. A triangular extended generalized quadrangle has the property that the number \(\mu\) of points collinear to two given non-collinear points is either 0 or \(\geq 2(t+1).\) The paper under review considers the case where \(\mu\) is 0 or exactly \(2(t+1)\) and the main result is that in this case every point of the generalized quadrangle \({\mathcal S}_ P\) is regular. Furthermore, if \(\mu\) is always \(2(t+1)\) (in which case \({\mathcal S}\) has a strongly regular point-graph), then the authors extend and improve an old result of \textit{F. Buekenhout} and \textit{X. Hubaut} [J. Algebra 45, 391-434 (1977; Zbl 0351.05021)]. However, there are no examples showing that still no further improvement can be obtained.
triangular extended generalized quadrangle
scientific article; zbMATH DE number 4190550