Projektivitätengruppen in angeordneten und topologischen Ebenen. (Groups of projectivities in ordered and topological planes)
type: Item
instance of: scholarly article
title: Projektivitätengruppen in angeordneten und topologischen Ebenen. (Groups of projectivities in ordered and topological planes)
zbMATH Open document ID: 0622.51007
DOI: 10.1007/BF01189867
published in: Archiv der Mathematik
publication date: 1986-01-01T00:00:00Z
review text: The influence of the structure of the group \(\Pi\) of all affine projectivities of a line onto itself or the group \(\Sigma\) of all projectivities of a line onto itself on the corresponding affine or projective plane has been studied for long. \textit{A. Schleiermacher} [Math. Z. 114, 313-320 (1970; Zbl 0182.231)] proved that \(\Pi\) or the stabilizer \(\Sigma_ p\) of a point p contains a regular normal subgroup if and only if the plane under consideration is a translation plane or a Moufang plane respectively. In a previous paper [Math. Z. 155, 11-21 (1977; Zbl 0354.50004)] the author deduced from this regularity condition that a locally compact connected affine plane is a translation plane if \(\Pi\) is locally compact in the compact-open topology (and hence \(\Pi\) is comparatively small). Weakening Schleiermacher's regularity condition or imposing topological conditions on the closure of \(\Pi\) in the group of all homeomorphisms of a line onto itself the author obtains various characterizations of desarguesian planes in the class of ordered planes and in the class of locally compact connected affine planes. In analogy to the description of the group \(\Sigma\) for projective Moulton planes of \textit{D. Betten} [J. Geom. 13, 197-209 (1979; Zbl 0426.51009)] the author shows in a short supplement that the group \(\Pi\) of an affine Moulton plane is isomorphic to the transformation group of all piecewise affine linear bijections of \({\mathbb{R}}\).
Mathematics Subject Classification ID: 51G05
Mathematics Subject Classification ID: 51H05
Mathematics Subject Classification ID: 51H10
zbMATH DE Number: 4009209
zbMATH Keywords: topological projective plane
zbMATH Keywords: ordered plane
zbMATH Keywords: affine projectivity
author: Karl Strambach
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
cites work: Topologies for Homeomorphism Groups
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cites work: Lacunary free actions on the real line
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cites work: Topological planes
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description: scientific article; zbMATH DE number 4009209