Semi-simple groups of compact 16-dimensional planes

Abstract

The automorphism group $\Sigma$ of a compact topological projective plane with a $16$-dimensional point space is a locally compact group. If the dimension of $\Sigma$ is at least $29$, then $\Sigma$ is known to be a Lie group. For the connected component $\Delta$ of $\Sigma$ the condition $\dim\Delta{\,\ge\,}27$ suffices. Depending on the structure of $\Delta$ and the configuration of the fixed elements of $\Delta$ sharper bounds are obtained here. Example: If $\Delta$ is semi-simple and fixes exactly one line and possibly several points on this line, then $\Delta$ is a Lie group if $\dim\Delta{\,\ge\,}11$. Any semi-simple group $\Delta$ which satisfies $\dim\Delta{\,\ge\,}25$ is a Lie group.