A FLAT LAGUERRE PLANE OF KLEINEWILLINGHÖFER TYPE V

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.

Flat Laguerre planes of Kleinewillinghöfer type E obtained by cut and paste

We provide examples of flat Laguerre planes of Kleinewillinghöfer type E, thus completing the classification of flat Laguerre planes with respect to Laguerre translations in B. Polster and G.F. Steinke, Results Maths. (2004). These planes are obtained by a method for constructing a new flat Laguerre plane from three given Laguerre planes devised in B. Polster and G. Steinke, Canad. Math. Bull. (1995) but no examples were given there.

The inner and outer space of 2-dimensional Laguerre planes

The classical 2-dimensional Laguerre plane is obtained as the geometry of non-trivial plane sections of a cylinder in R3 with a circle in R2 as base. Points and lines in R3 define subsets of the circle set of this geometry via the affine non-vertical planes that contain them. Furthemore, vertical lines and planes define partitions of the circle set via the points and affine non-vertical lines, respectively, contained in them.We investigate abstract counterparts of such sets of circles and partitions in arbitrary 2-dimensional Laguerre planes. We also prove a number of related results for generalized quadrangles associated with 2-dimensional Laguerre planes.

A Characterization of Some Geometries of Chains

The geometries considered here are the Möbius plane M() (W. Benz [1]), the Laguerre plane L() (W. Benz and H. Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G. Kaerlein [18]) over a field . All of them are geometries of an algebra with identity over a field. The characterization of the projective plane over a field by the proposition of Pappus first gave a close relation between algebraic and geometric structures. B. L. v. d. Waedern and L. J. Smid [28] presented a further example by characterizing the Möbius and Laguerre plane with incidence axioms and the "complete" proposition of Miquel.