A characterization of quaternion planes
There do exist locally compact stable planes \({\mathcal M}\) of topological dimension 2 (or 4) with \(SL_ 2{\mathbb{R}}\) (or \(SL_ 2{\mathbb{C}})\) as a group of automorphisms, such that \({\mathcal M}\) cannot be embedded into a projective plane [cf. \textit{K. Strambach}, Math. Ann. 179, 15-30 (1968; Zbl 0164.513) and \textit{R. Löwen}, Geom. Dedicata 21, 1-12 (1986; Zbl 0596.51007)]. On the contrary, the author shows that an 8-dimensional plane \({\mathcal M}\) of this kind can always be embedded into the quaternion plane if the automorphism group of \({\mathcal M}\) contains \(SL_ 2{\mathbb{H}}\). This is a corollary to the main theorem: \({\mathcal M}\) contains an open subplane which is isomorphic to the punctured affine quaternion plane and is left invariant by \(SL_ 2{\mathbb{H}}\), the action being equivalent to the natural one.
scientific article; zbMATH DE number 4175866