Flat Laguerre planes admitting 4-dimensional groups of automorphisms that fix a parallel class
type: Item
instance of: scholarly article
title: Flat Laguerre planes admitting 4-dimensional groups of automorphisms that fix a parallel class
zbMATH Open document ID: 1329.51007
author: Günter F. Steinke
published in: Aequationes Mathematicae
publication date: 2015-09-30T00:00:00Z
review text: The author classifies all 2-dimensional topological Laguerre planes whose automorphism group is 4-dimensional and fixes precisely one parallel class. Together with known results, mostly due to the author, this completes the classification of 2-dimensional Laguerre admitting an at least 4-dimensional group, except for the case that the group is transitive on the set of parallel classes but not on the point set. This last remaining case is not void; examples have been given by the author [Result. Math. 12, 207--221 (1987; Zbl 0632.51018)] and by the author together with the reviewer [J. Lie Theory 17, No. 4, 685--708 (2007; Zbl 1155.51008)]. To summarize the classification in the case where some parallel class is fixed, one may first say that all such planes belong to one of the following three types. Type (1). Consider a quadratic parabola \(y=ax^2\) in \(\mathbb R^2\). By taking as circles all translates of this curve for all nonzero values of \(a\) plus all ordinary lines we get a model of the classical Laguerre plane. Replacing \(x^2\) by a function of \(x\) whose graph sufficiently resembles a parabola, we can obtain non-classical Laguerre planes, called Laguerre planes of translation type. We may even take two distinct generating functions, depending on the sign of \(a\). Type (2). If, instead of translations \((x,y) \to (x+u,y+v)\), we use shears \((x,y) \to (x,y+kx+v)\), a similar procedure using a somewhat different class of fake parabolae yields the planes called Laguerre planes of shear type. Both types (1) and (2) were described by \textit{U. Pfüller} and the reviewer [Geom. Dedicata 23, 73--85 (1987; Zbl 0615.51006); ibid. 23, 87--96 (1987; Zbl 0615.51007)]. Type (3). The set of all translates of the parabola \(y = x^2\) together with the vertical lines forms a model of the real affine plane. Taking all primitives of the parabolae one gets a model of the classical Laguerre plane whose circles are graphs of degree 3 polynomials. Again, replacing \(x^2\) by…
reviewed by: Rainer Löwen
Mathematics Subject Classification ID: 51H15
Mathematics Subject Classification ID: 51B15
zbMATH DE Number: 6487920
zbMATH Keywords: Laguerre plane
zbMATH Keywords: topological incidence geometry
zbMATH Keywords: flat Laguerre plane
zbMATH Keywords: automorphism group
zbMATH Keywords: classification
MaRDI profile type: Publication
full work available at URL: s00010-014-0315-1
OpenAlex ID: W2074351050
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DOI: 10.1007/S00010-014-0315-1
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description: scientific article; zbMATH DE number 6487920