Simple connectedness of hyperplane complements in some geometries related to buildings
A gamma space is a point-line space in which, given any point-line pair \((p, L)\), the assumption \(|p^{\perp} \cap L| \geq 2\) implies \(L \subseteq p^{\perp}\), where \(p^{\perp}\) is the set of all the points collinear with \(p\). The geometries studied in this paper generalize polar geometries (and generalized quadrangles) that have been topics of intense study in the past. A parapolar space \(\Gamma\) is a connected space (in the sense of the underlying bipartite graph) which is a also a gamma space and has a certain family of convex subsets (defined in terms of geodesic paths in the underlying graph), called the symplecta. For a class of parapolar spaces that includes the geometries \(E_{6, 4}, E_{7, 7}\) and \(E_{8, 1}\) with lines of size at least three, the metasymplectic spaces with lines of size at least four, and the polar line Grassmannians with lines of size at least four with the exception of \(D_{4, 2}(3)\), the author shows that the subgraph of the point-line collinearity graph induced on the complement of a hyperplane is simply connected. The author also shows that parapolar spaces have Veldkamp lines.
scientific article; zbMATH DE number 5836895