Two-dimensional Laguerre planes with large automorphism groups
This paper is closely related to the preceding one. The automorphism group \(\Gamma\) of a non-Miquelian locally compact topological Laguerre plane \({\mathcal E}\) with a two-dimensional point set is a Lie group of dimension \(<6\). The authors give a complete and explicit classification of all planes \({\mathcal E}\) such that either \(\dim \Gamma =5\) or \(\Gamma\) fixes a point and \(\dim \Gamma =4.\) In the first case, the proper circles in an affine subplane are of the form \(y=af(x)+sx+v,\) where \(f(x)=x^{\alpha}\) for \(x\geq 0\), \(f(x)=\beta | x|^{\alpha}\) for \(x\leq 0\), \(\alpha >1\), \(\beta >0\) and \((\alpha,\beta)\neq (2,1):{\mathcal E}\) is of shear type over the skew parabola f. Similarly, the planes of the second sort are of translation type over a pair of skew parabolae.
scientific article; zbMATH DE number 3996511