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

2012 ◽  
Vol 84 (1-2) ◽  
pp. 99-119 ◽  
Author(s):  
Günter F. Steinke
Keyword(s):  
Laguerre Planes ◽  
Kleinewillinghöfer Type

scholarly journals A FLAT LAGUERRE PLANE OF KLEINEWILLINGHÖFER TYPE V

2011 ◽  
Vol 91 (2) ◽  
pp. 257-274 ◽  
Author(s):  
JEROEN SCHILLEWAERT ◽  
GÜNTER F. STEINKE
Keyword(s):  
Parameter Family ◽  
Laguerre Plane ◽  
The Other ◽  
Flat Laguerre Plane ◽  
Laguerre Planes ◽  
Type V ◽  
Central Automorphisms ◽  
Kleinewillinghöfer Type

AbstractThe 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.

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

2005 ◽  
Vol 72 (2) ◽  
pp. 213-223 ◽  
Author(s):  
Günter F. Steinke
Keyword(s):  
Laguerre Plane ◽  
Flat Laguerre Plane ◽  
Laguerre Planes ◽  
Kleinewillinghöfer Type

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.

A FAMILY OF TWO-DIMENSIONAL LAGUERRE PLANES OF KLEINEWILLINGHÖFER TYPE II.A.2

2018 ◽  
Vol 105 (3) ◽  
pp. 366-379
Author(s):  
GÜNTER F. STEINKE
Keyword(s):  
Two Dimensional ◽  
Transitive Groups ◽  
Laguerre Planes ◽  
Translation Type ◽  
Central Automorphisms ◽  
Existence Question ◽  
Kleinewillinghöfer Type

Kleinewillinghöfer classified Laguerre planes with respect to linearly transitive groups of central automorphisms. Polster and Steinke investigated two-dimensional Laguerre planes and their so-called Kleinewillinghöfer types. For some of the feasible types the existence question remained open. We provide examples of such planes of type II.A.2, which are based on certain two-dimensional Laguerre planes of translation type. With these models only one type, I.A.2, is left for which no two-dimensional Laguerre planes are known yet.

Flat Laguerre planes of Kleinewillinghöfer type III.B

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Jeroen Schillewaert ◽  
Günter F. Steinke
Keyword(s):  
Laguerre Planes ◽  
Kleinewillinghöfer Type

On the Kleinewillinghöfer types of flat Laguerre planes

2004 ◽  
Vol 46 (1-2) ◽  
pp. 103-122 ◽  
Author(s):  
Burkard Polster ◽  
Günter F. Steinke
Keyword(s):  
Laguerre Planes

scholarly journals The inner and outer space of 2-dimensional Laguerre planes

1997 ◽  
Vol 62 (1) ◽  
pp. 104-127 ◽  
Author(s):  
B. Polster ◽  
G. F. Steinke
Keyword(s):  
Outer Space ◽  
Laguerre Plane ◽  
Generalized Quadrangles ◽  
Base Points ◽  
Laguerre Planes

AbstractThe 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.

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