A Flat Laguerre Plane of Kleinewillinghöfer Type V

Abstract

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.

A Family of Two-Dimensional Laguerre Planes of Kleinewillinghöfer Type II.A.2

“…In [14] and [21], two-dimensional Laguerre planes were considered and their so-called Kleinewillinghöfer types were investigated, that is, the Kleinewillinghöfer types of the (full) automorphism groups. In particular, all feasible types of two-dimensional Laguerre planes with respect to Laguerre translations, were completely determined in [14], the case of Laguerre homotheties was dealt with in [21] and Laguerre homologies are covered in [14,17,22]; see Section 3 for definitions of these kinds of central Laguerre plane automorphisms. Examples for some of the feasible combined Kleinewillinghöfer types of twodimensional Laguerre planes (that is, with respect to all three types of central automorphisms Kleinewillinghöfer used in her classification) can be found in [14,Section 6], [9] and [20].…”

“…In two-dimensional Laguerre planes, 21 of these 46 combined types cannot occur. There are models of two-dimensional Laguerre planes of types I.A.1, I.B.1, I.B.3, I.C.1, I.E.1, I.E.4, I.G.1, I.H.1, I.H.11, II.A.1, II.E.1, II.E.4, II.G.1, III,B.1, III.B.3, III.H.1, III.H.11, IV.A.1, IV.A.2, V.A.1, VII.D.1, VII.D.8 and VII.K.13; see [14, Section 6],[9,16,17,[20][21][22]. Here a combined type just refers to the respective single types.…”