Semiclassical topological flat Laguerre planes obtained by pasting along two parallel classs
A topological flat Laguerre plane is called semiclassical if its point space is the union of two closed halfplanes whose induced subgeometries are both isomorphic to their classical counterparts. The intersection of these two halfplanes consists of either two parallel classes or one circle. Here the author investigates the first possibility whereas he discusses the second one in Result Mat. 12, 207-221 (1987; Zbl 0632.51018). Any semiclassical flat Laguerre plane of the first kind can be constructed with the aid of three orientation preserving homeomorphisms of \({\mathbb{R}}\) fixing 0 and vice versa. The author determines the isomorphism classes and the collineation groups of the planes under consideration. In most cases the automorphism group turns out to be 0-dimensional or even trivial but for suitably chosen generating functions its dimension can take on all possible values.
scientific article; zbMATH DE number 4078141