Compact generalised quadrangles with parameter 1 and large group of automorphisms
A compact generalized quadrangle is a generalized quadrangle such that the point and line spaces carry compact topologies and the incidence relation is closed. Even though there are restrictions on the topological dimensions (or parameters) of point rows and line pencils, infinitely many pairs occur. The situation changes drastically if these parameters are equal. In this case only the parameters \(0\), \(1\), \(2\) and possibly \(\infty\) occur, and in the two finite cases the point and line spaces are homeomorphic to \(\mathbb{P}_{3}(\mathbb{R})\) or \(\mathbb{P}_{3}(\mathbb{C})\) respectively. There is also a beautiful geometric implication: such a generalized quadrangle \(Q\) is anti-regular (up to duality) and as a consequence for every point \(p\) a Laguerre plane (called the daughter of \(Q\) at \(p\)) can be associated with \(Q\); furthermore this process can be reversed [\textit{A. E. Schroth}, Topological circle planes and topological quadrangles (Pitman Research Notes in Mathematics Series. 337. Longman) (1995; Zbl 0839.51013)]. In the parameter \(1\) case the collineation group of a daughter is a Lie group of dimension at most \(7\). All such daughters (which comprise all \(2\)-dimensional Laguerre planes) with an at least \(4\)-dimensional group have been classified by Steinke, but the result is partly unpublished. The author uses this classification to obtain related results about compact generalized quadrangles with parameter \(1\). In order to accomplish this task, Steinke's classification is described and investigated with respect to the sisterhood relation (i.e.\ two Laguerre planes are daughters of the same generalized quadrangle). This is useful, because two Laguerre planes are isomorphic if and only if the mothering generalized quadrangle admits a collineation mapping the two defining points onto each other. Using this information the author classifies all generalized quadrangles whose daughters have an at least \(4\)-dimensional collineation group. An application of this result yields that all generalized quadrangles with an at least \(6\)-dimensional group are classical, i.e.\ isomorphic to the real orthogonal or symplectic quadrangles, which have a \(10\)-dimensional group. The author mentions that the technique used in the paper does not lead to a classification of generalized quadrangles with a \(5\)-dimensional group, because a priori their daughters are not known to have an at least \(4\)-dimensional group.
scientific article; zbMATH DE number 1546673