Simple connectedness of hyperplane complements in some geometries related to buildings
type: Item
instance of: scholarly article
title: Simple connectedness of hyperplane complements in some geometries related to buildings
zbMATH Open document ID: 1253.51009
author: Anna Kasikova
published in: Journal of Combinatorial Theory. Series A
publication date: 2011-01-14T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51E99
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E24
zbMATH DE Number: 5836895
zbMATH Keywords: parapolar spaces
zbMATH Keywords: convex subspaces
zbMATH Keywords: symplecta
zbMATH Keywords: hyperplane complements
zbMATH Keywords: buildings
zbMATH Keywords: point-line space
reviewed by: Sharad Sane
MaRDI profile type: Publication
full work available at URL: j.jcta.2010.02.005
OpenAlex ID: W2005175823
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DOI: 10.1016/J.JCTA.2010.02.005
description: scientific article; zbMATH DE number 5836895