Four-dimensional compact projective planes with a nonsolvable automorphism group
Let \({\mathcal P}=(P,{\mathfrak L})\) be a non-arguesian topological projective plane having a compact point set P of topological dimension dim P\(=4\). Then P is in fact homeomorphic to the complex plane and the group \(\Sigma\) of continuous automorphisms of \({\mathcal P}\) is a Lie group with \(f=\dim \Sigma \leq 8.\) Theorem. If dim \(\Sigma\geq 7\), then \({\mathcal P}\) is a (dual) translation plane, or \({\mathcal P}\) is isomorphic to a unique shift plane due to N. Knarr. The case that \(\Sigma\) is solvable has been dealt with by \textit{D. Betten} [Geom. Dedicata 36, No.2, 151-170 (1990)] who has also explicitly determined all translation planes with \(f\geq 7\) [see \textit{D. Betten}, Math. Z. 154, 125-141 (1977; Zbl 0335.50016)]. Knarr's shift plane is given in \textit{D. Betten} and \textit{N. Knarr}, Abh. Math. Semin. Univ. Hamb. 57, 227-234 (1987; Zbl 0635.51011). The present author studies non-solvable groups \(\Sigma\) with \(f\geq 6\). If neither \({\mathcal P}\) nor its dual is a translation plane, then the connected component \(\Delta\) of \(\Sigma\) is isomorphic to the affine group \(\Lambda ={\mathbb{R}}^ 2GL^+_ 2{\mathbb{R}}\). The case where \(\Sigma\) contains a subgroup \(SO_ 3\) or \(Spin_ 3\) had been discussed earlier [the author, Monatsh. Math. 97, 55-61 (1984; Zbl 0526.51014) and Geom. Dedicata 21, 1-12 (1986; Zbl 0596.51007)]. In the only remaining case, a Levi complement \(\phi\) of the radical \(\sqrt{\Delta}\) is locally isomorphic to \(SL_ 2{\mathbb{R}}\). A detailed analysis shows that \(\Delta\) has trivial center, that \(\phi \cong SL_ 2{\mathbb{R}}\) and finally that \(\Delta\sim \Lambda\).
scientific article; zbMATH DE number 4171831