Four-dimensional compact projective planes with a solvable collineation group (Q1125234)

Each compact projective plane with point space of topological dimension \(4\) has an automorphism group \(\Sigma\) which is a Lie group. The planes with \(\dim\Sigma>6\) have been determined by \textit{D. Betten} [Geom. Dedicata 36, No. 2/3, 151-170 (1990; Zbl 0717.51013)] and \textit{R. Löwen} [Geom. Dedicata 36, No. 2/3, 225-234 (1990; Zbl 0712.51011)]. In the case \(\dim\Sigma=6\), the candidates for planes with nonsolvable \(\Sigma\) have been singled out by R. Löwen [op. cit.]. For \(\dim\Sigma<6\) any attempt at a classification appears hopeless: the group is too small to describe any substantial part of the flag space. The remaining case, where \(\Sigma\) is a solvable group of dimension~\(6\), is treated in the paper under review. It is shown that in each of these planes the group \(\Sigma\) fixes a line and acts doubly transitively on some parallel class in the corresponding affine plane. Thus the planes under consideration have been classified in [\textit{H. Klein}, Geom. Dedicata 61, No. 3, 227-255 (1996; Zbl 0861.51010)].