# Semi-simple groups of compact 16-dimensional planes

## TL;DR

This paper investigates the structure of automorphism groups of compact 16-dimensional projective planes, establishing conditions under which these groups are Lie groups, especially focusing on semi-simple groups and their fixed elements.

## Contribution

It provides new bounds on the dimensions of semi-simple automorphism groups that guarantee they are Lie groups, refining previous results based on fixed elements and group structure.

## Key findings

- Semi-simple groups fixing exactly one line are Lie groups if dimension ≥ 11.
- Semi-simple groups with dimension ≥ 25 are Lie groups.
- Sharper bounds depend on the fixed elements and group structure.

## 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.