A classification of 2-dimensional Laguerre planes admitting 3-dimensional groups of automorphisms in the kernel
The paper under review deals with the classification of all 2-dimensional, flat Laguerre planes with a big automorphism group. The point space of such a geometry is a cylinder \(S^1 \times \mathbb{R}\), and the circles are continuous functions \(S^1\to \mathbb{R}\). Steinke uses a construction of a large class of Laguerre planes due to Steinke himself and Polster. These planes depend on four functions \(f, g, h, k : \mathbb{R}\to \mathbb{R}\), and will be denoted by the symbol \(L(f,g,h,k)\). For some special functions \(f, g, h, k\), we obtain the so-called ovoidal Laguerre planes, i.e. Laguerre planes obtained from the cone over a oval in the real affine plane. In particular, we have the classical plane if the oval is a circle. The author considers the connected component \(\Delta\) of the group of all collineations fixing all parallel classes of our Laguerre plane. It is known that \(\Delta\) is solvable with \(\dim \Delta\leq 4\), and \(\dim\Delta = 4\) if and only if the plane is ovoidal. In the paper under review, all planes with \(\dim\Delta\geq 3\) are classified. The classification proceeds as follows. Assume \(\dim\Delta = 3\). First, if \(\Delta\simeq\mathbb{R}^3\) or if \(\Delta\) acts transitively on the set of all circles, then the plane is ovoidal and already known. The first bifurcation is whether \(\Delta\) fixes a point or not. If \(\Delta\) fixes a point, then the plane turns out to be of the form \(L(f,g,\text{id},1)\). Otherwise, \(\Delta\) acts transitively on each parallel class. Now, the connected component \(\Delta^1_\pi\) of the kernel of the action of \(\Delta\) on a parallel class \(\pi\) is studied. We have \(\dim \Delta^1_\pi\in \{1,2\}\), and if \(\dim \Delta^1_\pi = 2\) for some parallel class \(\pi\), then \(\Delta\) is transitive on the set of circles, and we are done. In the final case, we know \(\dim \Delta^1_\pi = 1\) for each parallel class \(\pi\). The group \(\Delta^1_\pi\) acts as a group of elations on an affine plane derived from our Laguerre plane. Either this group of elations consists of translations of the derived affine plane, or it is a group of shears. In the first case, the Laguerre plane becomes \(L(f,f,h,1)\) for some functions \(f\), \(h\), and in the second case we get a plane \(L(f,f,\text{id}, k)\) for suitable functions \(f\), \(k\).
scientific article; zbMATH DE number 1546662