Two-dimensional stable planes with non-solvable automorphism group
The author considers stable topological planes \({\mathcal P}\) whose point set \(P\) is a two-dimensional manifold, and he determines all such planes whose collineation group contains a semi-simple connected, closed subgroup \(\Delta\). Remarkably, it is not assumed that the lines are connected subsets of \(P\). In fact, under such additional connectedness assumptions, one could rely on the existing literature: all such planes having an at least 3-dimensional collineation group are explicitly known, and the planes with a semi-simple collineation group could be singled out from this long-existing classification, mainly worked out by \textit{H. Salzmann} [Adv. Math. 2, 1--60 (1967; Zbl 0153.21601)], \textit{K. Strambach} [Math. Ann. 179, 15--30 (1968; Zbl 0164.51302)] and \textit{D. Betten} [Math. Z. 107, 363--379 (1968; Zbl 0167.49002)]. In order to dispense with such extra connectedness assumptions, the author devises new methods allowing a direct and uniform treatment of the general case. Interestingly, the result is that entirely new planes do not arise and that the planes which are obtained additionally in the general case can all be imbedded as \(\Delta\)-invariant open subplanes in the planes which were already known from the former classification. As a starting point for the proofs, it is shown that \(\Delta\) has an open orbit \(U\) in \(P\). The classification of transitive Lie group actions as given by \textit{G. D. Mostow} [Ann. Math. (2) 52, 606--636 (1950; Zbl 0040.15204)] and, more recently and in completed form, by \textit{D. Betten} [Transitive Wirkungen auf Flächen, Math. Seminar Univ. Kiel (1977)] is then used to determine the possibilities for the action of \(\Delta\) on U and the geometry of \(U\). Finally, each case is checked for possible \(\Delta\)-equivariant extensions of \(U\) to larger geometries containing \(U\) as an open subset; here, the author can use his previous general investigations on equivariant embeddings of this kind [Monatsh. Math. 91, 19--37 (1981; Zbl 0453.51008)]. In further papers, the author intends to publish similar results on 4-dimensional stable planes, resulting in a classification of all 4-dimensional planes whose collineation group contains a semi-simple connected, closed subgroup not locally isomorphic to \(\mathrm{SL}_ 2(\mathbb R)\). The first paper in that series deals with projective planes admitting \(\mathrm{SO}_ 3(\mathbb R)\) as collineation group [Monatsh. Math. 97, 55--61 (1984; Zbl 0526.51014)].
scientific article; zbMATH DE number 3853882
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