Flat Laguerre planes admitting 4-dimensional groups of automorphisms that fix at least two parallel classes (Q2581049)

Locally compact connected Laguerre planes exist with point spaces of dimensions \(2\) and \(4\); the \(2\)-dimensional ones are also termed flat. Many (but not all) examples of flat Laguerre planes may be obtained by using the plane sections of the cone over an oval in the projective space over the reals. These examples are called ovoidal. The automorphism group \(\Gamma\) of a flat Laguerre plane \(\mathcal L\) is a Lie group of dimension at most~\(7\). The determination of all planes with \(\dim\Gamma\geq5\) has been completed by \textit{R. Löwen} and \textit{U. Pfüller} [Geom. Dedicata 23, 87--96 (1987; Zbl 0615.51007)]. Löwen and Pfüller [loc.\,cit.] have also solved the case where a subgroup \(\Sigma\leq\Gamma\) with \(\dim\Sigma\geq4\) fixes a point. A point-transitive action of a group of dimension at least~\(4\) is only possible on the classical plane [see \textit{G. F. Steinke}, Result. Math. 24, No. 3--4, 326--341 (1993; Zbl 0787.51012)]. The paper under review contains the classification of planes admitting \(\Sigma\leq\Gamma\) with \(\dim\Sigma=4\) such that \(\Sigma\) fixes at least two parallel classes. Apart from ovoidal planes, there also occur Laguerre planes of shear type [cf. \textit{B. Polster} and \textit{G. F. Steinke}, Bull. Aust. Math. Soc. 61, No. 1, 69--83 (2000; Zbl 0956.51006)]. The planes are described explicitly in the paper under review. Since flat Laguerre planes occur as Lie geometries of \(2\)-dimensional compact (anti\-regular) generalized quadrangles [\textit{A. E. Schroth}, Geom. Dedicata 36, No. 2/3, 365--373 (1990; Zbl 0717.51007)], the author's results also contribute to the theory of quadrangles.