Two-dimensional stable planes with non-solvable automorphism group
type: Item
instance of: scholarly article
title: Two-dimensional stable planes with non-solvable automorphism group
zbMATH Open document ID: 0537.51020
DOI: 10.1007/BF01198586
published in: Archiv der Mathematik
publication date: 1983-01-01T00:00:00Z
review text: The author considers stable topological planes \({\mathcal P}\) whose point set \(P\) is a two-dimensional manifold, and he determines all such planes whose collineation group contains a semi-simple connected, closed subgroup \(\Delta\). Remarkably, it is not assumed that the lines are connected subsets of \(P\). In fact, under such additional connectedness assumptions, one could rely on the existing literature: all such planes having an at least 3-dimensional collineation group are explicitly known, and the planes with a semi-simple collineation group could be singled out from this long-existing classification, mainly worked out by \textit{H. Salzmann} [Adv. Math. 2, 1--60 (1967; Zbl 0153.21601)], \textit{K. Strambach} [Math. Ann. 179, 15--30 (1968; Zbl 0164.51302)] and \textit{D. Betten} [Math. Z. 107, 363--379 (1968; Zbl 0167.49002)]. In order to dispense with such extra connectedness assumptions, the author devises new methods allowing a direct and uniform treatment of the general case. Interestingly, the result is that entirely new planes do not arise and that the planes which are obtained additionally in the general case can all be imbedded as \(\Delta\)-invariant open subplanes in the planes which were already known from the former classification. As a starting point for the proofs, it is shown that \(\Delta\) has an open orbit \(U\) in \(P\). The classification of transitive Lie group actions as given by \textit{G. D. Mostow} [Ann. Math. (2) 52, 606--636 (1950; Zbl 0040.15204)] and, more recently and in completed form, by \textit{D. Betten} [Transitive Wirkungen auf Flächen, Math. Seminar Univ. Kiel (1977)] is then used to determine the possibilities for the action of \(\Delta\) on U and the geometry of \(U\). Finally, each case is checked for possible \(\Delta\)-equivariant extensions of \(U\) to larger geometries containing \(U\) as an open subset; here, the author can use his previous general investigations on equivariant embeddings of this kind [Monatsh.…
reviewed by: Hermann Hähl
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 22A30
zbMATH DE Number: 3853882
zbMATH Keywords: stable topological planes
zbMATH Keywords: simple collineation groups
zbMATH Keywords: real hyperbolic plane
zbMATH Keywords: Moulton planes
zbMATH Keywords: transitive Lie group actions
author: Rainer Löwen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01198586
OpenAlex ID: W2068828604
description: scientific article; zbMATH DE number 3853882