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Journal of Lie Theory, Vol. 10, No. 1, pp. 127-146 (2000)
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Noncompact, almost simple groups operating on locally compact, connected translation planes

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Harald Löwe

Technische Universität Braunschweig

Institut für Analysis

Abteilung für Topologie und Grundlagen der Analysis

Pockelsstr. 14

38106 Braunschweig, Germany

harald@geometrie.tu-bs.de

**Abstract:** Let $\scriptstyle\Bbb{E}$ be a locally compact, connected translation plane. The aim of this paper is a detailed investigation of noncompact, connected, almost simple subgroups of the point stabilizer $G_0$ of $\Bbb E$. It turns out that the only possible groups $\scriptstyle\Delta$ of this kind are 2-fold covering groups of PSO$_m\scriptstyle({\Bbb R}, 1)$ for $\scriptstyle3\leq m\leq 10$. Moreover, the nontrivial central element of $\Delta$ is the reflection at 0. Furthermore, we will show the existence of an orbit ${\cal S}$ (called the {\eightit weight sphere}\/ of $\Delta$) homeomorphic to an $\scriptstyle(m-2)$-sphere on which the action of $\Delta$ is equivalent to the natural action of PSO$_m\scriptstyle({\Bbb R}, 1)$ on ${\Bbb S}_{m-2}$. This weight sphere ${\cal S}$ is characterized as the set of those lines in $\cal L_0$ whose stabilizer in $\Delta$ is a minimal parabolic subgroup of $\Delta$. As a by-product we prove that a semisimple subgroup of $G_0$ always has real rank 0 or 1.

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