Solvable Collineation Groups of Smooth Projective Planes

Abstract: Let $ \Delta$ be a closed solvable subgroup of the collineation group of one of the four classical Moufang planes ${\cal P}_2 F$ over the domain $ F\in\{\hbox R, C, H, O\}$, where $\bf H$ denotes the quaternions and $\bf O$ the Cayley numbers. Then $ \dim \Delta\leq4,9,17,30$ if $ F=\hbox R, C, H, O$. For arbitrary compact connected projective planes $\cal P$ of finite dimension $n$ M. Lüneburg derived upper bounds for the dimension of $ \Delta$ depending on $n$ and the configuration of the fixed elements of $ \Delta$, see [ML]. In this paper we consider $ \dim \Delta$ in the case of smooth projective planes. Compared to the results of [ML], the upper bounds of $ \dim \Delta$ can be lowered in almost every case; except for one single case we even derive sharp bounds.

[ML] Lüneburg, M.: Involutionen, auflösbare Gruppen und die Klassifikation topologischer Ebenen. Mitt. Math. Sem. Giessen 209 (1992), 1-113

Keywords: Moufang planes; collineation group; projective planes

Classification (MSC91): 51A35

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