A family of flat Laguerre planes of Kleinewillinghöfer type IV.A

Abstract

Just like Lenz–Barlotti classes reflect transitivity properties of certain groups of central collineations in projective planes, Kleinewillinghofer 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 Kleinewillinghofer types with respect to Laguerre homologies and also with respect to Laguerre homotheties are now accounted for, and the number of open cases of Kleinewillinghofer types (with respect to Laguerre homologies, Laguerre translations and Laguerre homotheties combined) is reduced to two.