SU4 (?) als Kollineationsgruppe in sechzehndimensionalen lokalkompakten Translationsebenen

Abstract

We study 16-dimensional locally compact translation planes in which, for an affine point o, the stabilizer \(\mathbb{G}_o \) of the affine collineation group \(\mathbb{G}\) contains a subgroup Σ locally isomorphic to SU4 (ℂ). If ⌆ has only one affine fixed point o, then it is shown that either the plane is the classical Moufang plane over the Cayley numbers, or else Σ must be normal in the stabilizer \(\mathbb{G}_o \) and \(\mathbb{G}\) has dimension at most 37. This also comprises the proof of the fact that if \(\mathbb{G}_o \) contains a subgroup locally isomorphic to SU4(ℂ) × SL2(ℂ) then the plane is the classical Cayley plane. The case that ⌆ has more affine fixed points in dealt with as well; then, except for a well-known family of planes admitting Spin7(ℝ) as a group of collineations, \(\mathbb{G}\) has dimension at most 34.