Geometries on Surfaces

A note on Laguerre translations

“…The aim of this paper is to use the close relationship between 2n-dimensional Laguerre planes for n = 1, 2 and compact antiregular generalized quadrangles with parameter n and give a partial answer to Problem 5.9.8 in [7] in that we show that in fact for an automorphism of such a Laguerre plane L not in the kernel of L the translation property at a point suffices in order to have a Laguerre translation. (The kernel of a Laguerre plane consists of all automorphisms that fix each parallel class and is a normal subgroup of the automorphism group.)…”

“…and elements of C are, considered as subsets of P , each homeomorphic to the n-sphere S n . For more information about topological Laguerre planes we refer to [1], [2], [7] and [10].…”

Divisible designs, Laguerre geometry, and beyond

“…The investigation of topological circle planes is part of the book [98]. It contains a wealth of bibliographical data.…”

A Flat Laguerre Plane of Kleinewillinghöfer Type V

“…A flat Laguerre plane or topological, locally compact, two-dimensional Laguerre plane L is an incidence structure of points and circles whose point set Z is the cylinder S 1 × R and whose circles C ∈ C are graphs of continuous functions from S 1 to R such that any three points, no two of which lie on the same generator {c} × R of the cylinder, can be joined by a unique circle and also such that the circles that touch a fixed circle K at p ∈ K partition the complement in Z of the generator that contains p. For more information on flat Laguerre planes, we refer to [2,3] or [10,Ch. 5].…”