Characterization of 16-dimensional Hughes planes (Q1270258)

Mirroring a defining property of the well-known finite Hughes planes, a compact 16-dimensional topological projective plane \({\mathcal P}\) is called a Hughes plane, if it admits a subplane \({\mathcal F}\) which is isomorphic to the classical quaternion plane, invariant under the automorphism group \(\Sigma\) of \({\mathcal P}\), and where \(\Sigma\) induces the full automorphism group \(\text{PSL}_3 {\mathbf H}\) of \({\mathcal F}\). Supplementing the classification of compact (connected) planes with an automorphism group of sufficiently large dimension [see the book of \textit{H. Salzmann} and his pupils, `Compact projective planes', de Gruyter, Berlin, New York (1996; Zbl 0851.51003)], the author gives the following characterization of these 16-dimensional Hughes planes. ``Assume that the compact projective plane \({\mathcal P}\) is not isomorphic to the classical Moufang plane over the octonions. Let \(\Delta\) be a closed subgroup of \(\Aut {\mathcal P}\). If \(\dim\Delta\geq 31\) and if \(\Delta\) has a normal torus subgroup, then \({\mathcal P}\) is a Hughes plane, \(\Delta=\Aut{\mathcal P}\), and \(\Delta' =\text{PSL}_3 {\mathbf H}\)''.