Four-dimensional compact projective planes with collineation group \(N_{6,28}\)
This paper deals with the last remaining case in the classification of all flexible 4-dimensional compact projective planes, i.e. topological projective planes with point space homeomorphic to the classical plane over the complex numbers such that the group of continuous collineations has an open orbit in the space of flags. Flexible planes have collineation groups of dimension at least 6. All 4-dimensional compact projective planes admitting a 7-dimensional collineation group have been classified. Up to duality these planes are translation planes or the shift plane of Knarr. Flexible 4-dimensional compact projective planes with 6-dimensional collineation groups have been classified by a number of authors except for one case, which is studied in this paper. It is assumed that the connected component \(\Sigma\) of the group of continuous collineations contains no subgroup isomorphic to the vector group \(\mathbb{R}^4\) and that \(\Sigma\) is solvable. In this case \(\Sigma\) fixes a flag \((v,W)\). It is then further assumed that, up to duality, \(\Sigma\) acts transitively on the line pencil \({\mathcal L}_v \backslash \{W\}\) and fixes no point on \(W\) except \(v\). The author [Geom. Dedicata 58, No. 1, 53-62 (1995); see the paper below] proved under these assumptions that the nilradical \(N\) of \(\Sigma\), i.e. the maximal connected nilpotent invariant subgroup of \(\Sigma\), is at least four-dimensional. Only the case of four-dimensional nilradicals has not been treated yet. The author proves that under the above assumptions the Lie algebra of \(N\) is isomorphic to the direct sum of a one-dimensional real Lie algebra and the three-dimensional nonabelian nilpotent real Lie algebra \(\langle x,y,z |[x,y] = z \rangle\). Models for planes admitting this type of collineation groups exist. Up to isomorphism there are precisely two four-dimensional nonabelian nilpotent real Lie algebras. One such Lie algebra is the one described above, the other Lie algebra, \({\mathcal N}\), is the extension of \(\mathbb{R}^3\) by the endomorphism \((x,y,z) \mapsto (- y,z, 0)\). There is exactly one six-dimensional solvable Lie algebra, called \(N_{6,28}\), containing \({\mathcal N}\) as its nilradical. To prove his result the author assumes that \(\Sigma\) has Lie algebra \(N_{6,28}\). He then uses the subalgebra structure of \(N_{6,28}\) and analyses the possible orbit structures of \(\Sigma\) on \({\mathcal L}_v \backslash \{W\}\) and on the affine plane obtained by removing \(W\). In each of the various cases a contradiction is eventually obtained and Lie algebra \(N_{6,28}\) is excluded.
scientific article; zbMATH DE number 792561
scientific article