Four-dimensional compact projective planes with two fixed points
type: Item
instance of: scholarly article
title: Four-dimensional compact projective planes with two fixed points
zbMATH Open document ID: 0849.51007
DOI: 10.1007/BF01223032
author: Dieter Betten
author: Hauke Klein
published in: Journal of Geometry
publication date: 1996-05-01T00:00:00Z
review text: The paper under review studies topological projective planes whose point space is a four-dimensional topological manifold. Endowed with the compact-open topology, the connected component \(\Sigma\) of the group of all continuous collineations of such a plane is a Lie group acting continuously on the space of points, lines and flags. Of particular interest is the case where \(\Sigma\) has an open orbit in the space of flags, which allows to reconstruct the plane from the given action of the group. In this case, one has \(\dim \Sigma \geq 6\). The authors assume in addition that \(\Sigma\) is solvable, fixes two points, and acts transitively on the complement of the joining line in both of the corresponding line pencils. Using the classification of solvable Lie algebras of low dimension (via their nilpotent radical), the authors prove that these assumptions characterize a family of planes which can be described by a single construction, involving two real parameters. Except for a single pair of parameters, these planes appear to be new examples of four-dimensional compact projective planes.
Mathematics Subject Classification ID: 51H10
zbMATH DE Number: 872714
zbMATH Keywords: two fixed points
zbMATH Keywords: four-dimensional compact projective planes
reviewed by: Markus Stroppel
MaRDI profile type: Publication
cites work: 4-dimensionale Translationsebenen mit 8-dimensionaler Kollineationsgruppe
cites work: Komplexe Schiefparabel-Ebenen
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full work available at URL: https://doi.org/10.1007/bf01223032
OpenAlex ID: W1981570400
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description: scientific article; zbMATH DE number 872714