4-dimensional point-transitive groups of automorphisms of 2-dimensional Laguerre planes
The main result states that a 2-dimensional Laguerre plane that admits a closed connected 4-dimensional point-transitive group of automorphisms is already classical. Furthermore, it is proved that up to conjugacy in the automorphism group of the classical 2-dimensional Laguerre plane there are only two different closed connected 4-dimensional point-transitive subgroups. The paper is a major contribution to the classification of 2-dimensional Laguerre planes with a large automorphism group. More precisely, all 2- dimensional Laguerre planes that admit an automorphism group of at least 5 are already classified [\textit{R. Löwen} and \textit{U. Pfüller}, Geom. Dedicata 23, 87-96 (1987; Zbl 0615.51007)], so the paper is the first step towards classifying all 2-dimensional Laguerre planes that admit an automorphism group of at least 4. Since 2-dimensional Laguerre planes are closely related to 3-dimensional generalised quadrangles [the reviewer, Geom. Dedicata 36, No. 2/3, 365- 373 (1990; Zbl 0717.51007)], the paper contributes also to the classification of 3-dimensional generalised quadrangles. This contribution is twofold. The results as such can be used directly but also the techniques developed in the proofs will most likely be suitable for 3-dimensional generalised quadrangles as well.
scientific article; zbMATH DE number 484147
scientific article