Flat Laguerre planes admitting 4-dimensional groups of automorphisms that fix a parallel class
The author classifies all 2-dimensional topological Laguerre planes whose automorphism group is 4-dimensional and fixes precisely one parallel class. Together with known results, mostly due to the author, this completes the classification of 2-dimensional Laguerre admitting an at least 4-dimensional group, except for the case that the group is transitive on the set of parallel classes but not on the point set. This last remaining case is not void; examples have been given by the author [Result. Math. 12, 207--221 (1987; Zbl 0632.51018)] and by the author together with the reviewer [J. Lie Theory 17, No. 4, 685--708 (2007; Zbl 1155.51008)]. To summarize the classification in the case where some parallel class is fixed, one may first say that all such planes belong to one of the following three types. Type (1). Consider a quadratic parabola \(y=ax^2\) in \(\mathbb R^2\). By taking as circles all translates of this curve for all nonzero values of \(a\) plus all ordinary lines we get a model of the classical Laguerre plane. Replacing \(x^2\) by a function of \(x\) whose graph sufficiently resembles a parabola, we can obtain non-classical Laguerre planes, called Laguerre planes of translation type. We may even take two distinct generating functions, depending on the sign of \(a\). Type (2). If, instead of translations \((x,y) \to (x+u,y+v)\), we use shears \((x,y) \to (x,y+kx+v)\), a similar procedure using a somewhat different class of fake parabolae yields the planes called Laguerre planes of shear type. Both types (1) and (2) were described by \textit{U. Pfüller} and the reviewer [Geom. Dedicata 23, 73--85 (1987; Zbl 0615.51006); ibid. 23, 87--96 (1987; Zbl 0615.51007)]. Type (3). The set of all translates of the parabola \(y = x^2\) together with the vertical lines forms a model of the real affine plane. Taking all primitives of the parabolae one gets a model of the classical Laguerre plane whose circles are graphs of degree 3 polynomials. Again, replacing \(x^2\) by a function whose graph resembles a parabola in a suitable sense (slightly different from the above), one obtains nonclassical Laguerre planes, called Laguerre planes of shift type. The construction is due to \textit{B. Polster} [Arch. Math. 64, No. 1, 75--85 (1995; Zbl 0813.51007)]. The planes of the above types have a 4-dimensional group only for special choices of generating functions; we shall not go into the details here. For the proof of the main result, the group is used to derive some functional equations for the functions whose graphs form the circles of the Laguerre plane. The hard part is to solve these functional equations.
scientific article; zbMATH DE number 6487920