Q1084338
type: Item
instance of: scholarly article
title: Achtdimensionale lokalkompakte Translationsebenen mit mindestens 17- dimensionaler Kollineationsgruppe. (Eight-dimensional locally compact translation planes with at least 17-dimensionally collineation group)
zbMATH Open document ID: 0605.51011
DOI: 10.1007/BF00181535
published in: Geometriae Dedicata
publication date: 1986-01-01T00:00:00Z
review text: Let \({\mathcal P}\) be a (locally) compact connected projective plane of finite topological dimension; only dimensions 2,4,8, and 16 are possible by a deep result of \textit{R. Löwen} [J. Reine Angew. Math. 343, 108-122 (1983; Zbl 0524.57011)]. The most interesting planes are those with large automorphism group; largeness may be measured by the topological dimension of the group \(\Sigma\) of all continuous collineations of \({\mathcal P}\). By results of \textit{H. Salzmann} [yet unpublished, cp. his paper in Math. Z. 176, 345-357 (1981; Zbl 0465.51004)], if dim \({\mathcal P}=8\) and dim \(\Sigma\geq 17\), then \({\mathcal P}\) is a translation plane or a dual translation plane or one of the Hughes planes described in \textit{H. Salzmann} [loc. cit.]. The paper under review concludes Hähl's classification, announced in the author's paper in Math. Z. 159, 259-294 (1978; Zbl 0381.51009), of all locally compact translation planes \({\mathcal P}\) of dimension 8 with dim \(\Sigma\geq 17\). The result may be summarized as follows. If dim \(\Sigma\geq 19\), then \({\mathcal P}\) is the desarguesian plane over the quaternions \({\mathbb{H}}\); there are several (explicitly known) families of nondesarguesian planes with dim \(\Sigma\in \{17,18\}\), see (b)-(f) below. We remark that there are analogous results of Salzmann for the other possible dimensions of \({\mathcal P}\), again leading to translation planes, as well as classifications of these translation planes, see \textit{D. Betten} [Math. Z. 154, 125-141 (1977; Zbl 0335.50016), Geom. Dedicata 16, 179-193 (1984; Zbl 0547.51012)] and the author [loc. cit.; Arch. Math. 48, 267-276 (1987); ''Eine Kennzeichnung der Oktavenebene'', Indag. Math., to appear; ''16-dimensionale lokalkompakte Translationsebenen mit SU(4,\({\mathbb{C}})\) als Kollineationsgruppe'', Geom. Dedicata, to appear]. The author contructs a large class of quasifields by modifying the multiplication of \({\mathbb{H}}:\) let \(h\in {\mathbb{H}}\) and le…
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 12K10
zbMATH DE Number: 3977826
zbMATH Keywords: topological projective plane
zbMATH Keywords: translation plane
zbMATH Keywords: quasifields
zbMATH Keywords: division algebra
author: Hermann Hähl
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00181535
OpenAlex ID: W1494643436
label: Achtdimensionale lokalkompakte Translationsebenen mit mindestens 17- dimensionaler Kollineationsgruppe. (Eight-dimensional locally compact translation planes with at least 17-dimensionally collineation group)
description: scientific article; zbMATH DE number 3977826