Journal of Lie Theory, Vol. 10, No. 1, pp. 127-146 (2000)

Noncompact, almost simple groups operating on locally compact, connected translation planes

Harald Löwe

Technische Universität Braunschweig
Institut für Analysis
Abteilung für Topologie und Grundlagen der Analysis
Pockelsstr. 14
38106 Braunschweig, Germany

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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