Actions of \(Spin_ 3\) on 4-dimensional stable planes
The author considers stable planes \({\mathcal E}\) with point space M which is assumed to be locally compact and Hausdorff and of covering dimension 4. The automorphism group \(\Gamma\) of \({\mathcal E}\) is locally compact, and if \(\Gamma\) is at least 5-dimensional, then it is a Lie group. Using the structure of Lie groups and the geometric structure one starts to classify stable planes with respect to the dimension of their automorphism group \(\Gamma\), i.e. for given integer f one asks for all stable planes with dim \(\Gamma\geq f\). This was carried out by the author for \(f=10\) [Geom. Dedicata 5, 239-294 (1976; Zbl 0344.50003)] and even for \(f=9\) (unpublished). However, if one requires that the automorphism group is not solvable, much smaller bounds f suffice. In this sense the author shows in this note that a stable plane whose automorphism group is at least 6- dimensional and which contains a closed connected subgroup isomorphic to Spin\({}_ 3\) is one of the following nine planes: an open subplane of the classical complex projective plane \({\mathcal P}_ 2{\mathbb{C}}\) which is the complement of \(\emptyset\), a point, a line, and an antiflag \(p\not\in L\), respectively, the three hyperbolic planes (interior, exterior, united), and the two \(SL_ 2{\mathbb{C}}\)-planes \({\mathcal S}_{{\mathbb{C}}}\) and \({\mathcal S}_{{\mathbb{C}}}\) minus the origin (\({\mathcal S}_{{\mathbb{C}}}\) is a 4- dimensional analogue to Strambach's 2-dimensional \(SL_ 2{\mathbb{R}}\)-plane and is constructed here). Combining this with the results of a series of previous papers of the author all stable planes whose automorphism group contains an at least 4- dimensional semisimple closed connected subgroup are determined. Beside the nine stable planes already mentioned there are two further planes which are the complements in \({\mathcal P}_ 2{\mathbb{C}}\) of the real projective subplane \({\mathcal P}_ 2{\mathbb{R}}\) and of a conic section, respectively. 2-dimensional stable planes admitting a semisimple closed connected subgroup have already been completely classified by the author [Arch. Math. 41, 565-571 (1983; Zbl 0537.51020)].
scientific article; zbMATH DE number 3960408