The classification of Kleinewillinghöfer types of \(2\)-dimensional Laguerre planes
type: Item
instance of: scholarly article
title: The classification of Kleinewillinghöfer types of \(2\)-dimensional Laguerre planes
zbMATH Open document ID: 1476.51011
published in: Aequationes Mathematicae
publication date: 2021-09-23T00:00:00Z
review text: For 2-dimensional Laguerre planes only 25 Kleinewillinghöfer classes with respect to central automorphisms are possible, 24 of which have been shown to be non-void, [the author, J. Aust. Math. Soc. 105, 366--379 (2018; Zbl 1406.51003)]. The author continues his work and presents a family \(\mathcal{L}(q)\), \(q>0\) of models for the remaining class I.A.2., thus affirming the existence of models for all possible types of the classification. Real-valued skew parabola functions \(f_q\), \(q > 0\) where \(f_q(x):= x^2\), if \(x\geq 0\), \(f_q(x):=qx^2\), if \(x < 0\) give rise to Laguerre planes of type IV.A.1 from [the author, Aequationes Math. 84, 99--119 (2012; Zbl 1271.51002)]. Using a cut-and-paste procedure from [\textit{B. Polster} and the author, Can. Math. Bull. 38, 469--480 (1995; Zbl 0844.51005)], the author gets a family \(\mathcal{L}(q), q > 0\) of 2-dimensional Laguerre planes. He gives a detailed description of the automorphism group of \(\mathcal{L}(q)\), which allows to infer that \(\mathcal{L}(q)\) is of type I.A.2.
Mathematics Subject Classification ID: 51H15
Mathematics Subject Classification ID: 51B15
zbMATH DE Number: 7399553
zbMATH Keywords: Laguerre plane
zbMATH Keywords: topological incidence geometry
zbMATH Keywords: \(2\)-dimensional Laguerre plane
zbMATH Keywords: central automorphism
zbMATH Keywords: Kleinewillinghöfer type
author: Günter F. Steinke
reviewed by: Horst Szambien
MaRDI profile type: Publication
full work available at URL: s00010-021-00780-3
OpenAlex ID: W3126232390
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DOI: 10.1007/S00010-021-00780-3
description: scientific article; zbMATH DE number 7399553