16-dimensional compact projective planes with a large group fixing two points and only one line

Abstract

We complete the determination of all pairs $\left(\mathcal{P},\Delta \right)$, where $\mathcal{P}$ is a compact projective plane with a $16$-dimensional point set, $\mathrm{\Delta }$ is an automorphism group of $\mathcal{P}$ of dimension at least $35$, and $\mathrm{\Delta }$ does not fix exactly one point and one line. If $\mathrm{\Delta }$ fixes two points and only one line, then $\mathrm{\Delta }$ contains a $15$-dimensional translation group and a compact subgroup ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_7ℝ$; hence $dim\mathrm{\Delta }\ge 36$. The planes are described by their coordinatizing Cartesian fields, more explicitly for $dim\mathrm{\Delta }>36$.