On near hexagons and spreads of generalized quadrangles
A generalized quadrangle is a connected partial linear space for which the diameter of the point graph \(\Gamma\) is two and such that for every point \(p\) and every line \(L\), there is a unique point \(q\) on \(L\) nearest to \(p\). And similarly, a near hexagon is a connected partial linear space for which the diameter of the point graph \(\Gamma\) is three and such that for every point \(p\) and every line \(L\), there is a unique point \(q\) on \(L\) nearest to \(p\). A spread of a generalized quadrangle is a set of lines which partitions the point set of the generalized quadrangle. The above mentioned incidence structures can be linked to each other since generalized quadrangles can appear as geodetically closed subsets within near hexagons. The author makes use of this relation between generalized quadrangles and near hexagons by presenting a glueing-construction, making use of two generalized quadrangles with a spread in each of them, which yields under some conditions a near hexagon. These latter hexagons are called glued near hexagons. This construction is applied to different generalized quadrangles having a spread, leading to new examples of near hexagons. Also the properties of these glued near hexagons are investigated. This information on the properties is then used to characterize the glued near hexagons within the set of all near hexagons.
scientific article; zbMATH DE number 1495745