Abstract
Just like Lenz–Barlotti classes reflect transitivity properties of certain groups of central collineations in projective planes, Kleinewillinghöfer types reflect transitivity properties of certain groups of central automorphisms in Laguerre planes. In the case of flat Laguerre planes, Polster and Steinke have shown that some of the conceivable types cannot exist, and they gave models for most of the other types. Only few types are still in doubt. Two of them are types IV.A.1 and IV.A.2, whose existence we prove here. In order to construct our models, we make systematic use of the restrictions imposed by the group generated by all central automorphisms guaranteed in type IV. With these models all simple Kleinewillinghöfer types with respect to Laguerre homologies and also with respect to Laguerre homotheties are now accounted for, and the number of open cases of Kleinewillinghöfer types (with respect to Laguerre homologies, Laguerre translations and Laguerre homotheties combined) is reduced to two.
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References
Groh H.: Topologische Laguerreebenen I. Abh. Math. Sem. Univ. Hamburg 32, 216–231 (1968)
Groh H.: Characterization of ovoidal Laguerre planes. Arch. Math. 20, 219–224 (1969)
Groh H.: Topologische Laguerreebenen II. Abh. Math. Sem. Univ. Hamburg 34, 11–21 (1969/1970)
Kleinewillinghöfer, R.: Eine Klassifikation der Laguerre-Ebenen. PhD thesis, Technische Hochschule Darmstadt (1979)
Kleinewillinghöfer R.: Eine Klassifikation der Laguerre-Ebenen nach \({\mathcal{L}}\) -Streckungen und \({\mathcal{L}}\) -Translationen. Arch. Math. 34, 469–480 (1980)
Löwen R., Pfüller U.: Two-dimensional Laguerre planes with large automorphism groups. Geom. Dedicata 23, 87–96 (1987)
Löwen R., Steinke G.F.: Actions of \({\mathbb{R}\cdot{\widetilde{\text{SL}}_2\mathbb{R}}}\) on Laguerre planes related to the Moulton planes. J. Lie Th. 17, 685–708 (2007)
Pickert G.: Projektive Ebenen. 2nd edn. Springer, Berlin (1975)
Polster B., Steinke G.F.: Criteria for two-dimensional circle planes. Beitr. Algebra Geom. 35, 181–191 (1994)
Polster, B., Steinke, G.F.: Geometries on surfaces. Encyclopedia of Mathematics and its Applications, vol. 84. Cambridge University Press, Cambridge (2001)
Polster B., Steinke G.F.: On the Kleinewillinghöfer types of flat Laguerre planes. Result. Math. 46, 103–122 (2004)
Salzmann H., Betten D., Grundhöfer T., Hähl H., Löwen R., Stroppel M.: Compact Projective Planes. de Gruyter, Berlin (1995)
Schillewaert J., Steinke G.F.: Flat Laguerre planes of Kleinewillinghöfer type III. B. Adv. Geom. 11, 637–652 (2011)
Schillewaert J., Steinke G.F.: A flat Laguerre plane of Kleinewillinghöfer type V. J. Austr. Math. Soc. 91, 257–274 (2011)
Steinke G.F.: On the structure of the automorphism group of 2-dimensional Laguerre planes. Geom. Dedicata 36, 389–404 (1990)
Steinke G.F.: A note on Laguerre translations. Innov. Incidence Geom. 2, 93–100 (2005)
Steinke G.F.: More on Kleinewillinghöfer types of flat Laguerre planes. Result. Math. 51, 111–126 (2007)
Wang X.: A simple proof of Descartes’s rule of sign. Am. Math. Monthly 111, 525–526 (2004)
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Steinke, G.F. A family of flat Laguerre planes of Kleinewillinghöfer type IV.A. Aequat. Math. 84, 99–119 (2012). https://doi.org/10.1007/s00010-011-0111-0
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DOI: https://doi.org/10.1007/s00010-011-0111-0