On the Kleinewillinghöfer types of flat Laguerre planes

Polster, Burkard; Steinke, Günter F.

doi:10.1007/bf03322874

Cited by 10 publications

(39 citation statements)

References 18 publications

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“…Hence, the set Z of all circles C for which the automorphism group of L is linearly transitive with respect to C-homologies contains a flock of circles. By [11,Proposition 3.4], the only set Z containing a flock of circles and one extra circle is of type VII and such a flat Laguerre plane is ovoidal by [11,Corollary 3.2]. However, the derived plane A (∞,0) at (∞, 0) is not Desarguesian by Proposition 5.4, and we have reached a contradiction.…”

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confidence: 87%

“…The centre of the induced homology is the point ω at infinity corresponding to the parallel classes of L. Kleinewillinghöfer [5, Satz 3.1] found seven types of groups of automorphisms of Laguerre planes, labelled I, II, III, IV, V, VI and VII with respect to Laguerre homologies. It is known that type VI cannot occur as the type of a flat Laguerre plane (see [11,Proposition 3.4]). The [5] A Laguerre homologies has group dimension two or three.…”

Section: Kleinewillinghöfer Types Of Flat Laguerre Planesmentioning

confidence: 99%

“…Polster and Steinke [11] and Steinke [16] considered flat Laguerre planes and investigated their so-called Kleinewillinghöfer types, that is, the Kleinewillinghöfer types of their full automorphism groups. In particular, all possible types of flat Laguerre planes with respect to Laguerre translations (automorphisms of Laguerre planes that either are the identity or fix the points of precisely one parallel class and induce a translation in the derived affine plane at one of its fixed points) were completely determined in [11]. The case of Laguerre homotheties (automorphisms of Laguerre planes that either are the identity or fix precisely two nonparallel points and induce a homothety in the derived affine plane at each of these two fixed points) was dealt with in [16].…”

Section: Introductionmentioning

confidence: 99%

“…https://doi.org/10.1017/S1446788711001534 [11] A flat Laguerre plane of Kleinewillinghöfer type V 267…”

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confidence: 99%

“…However, the derived plane A (∞,0) at (∞, 0) is not Desarguesian by Proposition 5.4, and we have reached a contradiction. Hence, L must be of Kleinewillinghöfer type V. It follows from the list of Kleinewillinghöfer types given in [11,Theorem 6.1] that L is of type V.A.1.…”

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confidence: 99%

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A Flat Laguerre Plane of Kleinewillinghöfer Type V

Schillewaert

Steinke

2011

J. Aust. Math. Soc.

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The Kleinewillinghöfer types of Laguerre planes reflect the transitivity properties of certain groups of central automorphisms. Polster and Steinke have shown that some of the conceivable types for flat Laguerre planes cannot exist and given models for most of the other types. The existence of only a few types is still in doubt. One of these is type V.A.1, whose existence we prove here. In order to construct our model, we make systematic use of the restrictions imposed by the group. We conjecture that our example belongs to a one-parameter family of planes all of type V.A.1.

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confidence: 87%

Section: Kleinewillinghöfer Types Of Flat Laguerre Planesmentioning

confidence: 99%