Groups of compact 8-dimensional planes: conditions implying the Lie property

Abstract

The automorphism group $\mathrm{\Sigma }$ of a compact topological projective plane with an $8$-dimensional point space is a locally compact group. If the dimension of $\mathrm{\Sigma }$ is at least $12$, then $\mathrm{\Sigma }$ is known to be a Lie group. For the connected component $\mathrm{\Delta }$ of $\mathrm{\Sigma }$ it is shown that $dim\mathrm{\Delta }\ge 10$ suffices, if $\mathrm{\Delta }$ is semisimple or does not fix exactly a nonincident point-line pair or a double-flag. $\mathrm{\Delta }$ is also a Lie group, if $\mathrm{\Delta }$ has a compact connected $1$-dimensional normal subgroup and $dim\mathrm{\Delta }\ge 11$.