
Journal of Lie Theory 18 (2008), No. 4, 961978 Copyright Heldermann Verlag 2008 Proper Actions on CorankOne Reductive Homogeneous Spaces Fanny Kassel Dép. de Mathématiques, Bâtiment 425, Faculté des Sciences, Université ParisSud 11, 91405 Orsay, France fanny.kassel@math.upsud.fr [Abstractpdf] \def\kkk{{\bf k}} Let $\kkk$ be a local field, $G$ the set of $\kkk$points of a connected semisimple algebraic $\kkk$group $\bf G$, and $H$ the set of $\kkk$points of a connected reductive algebraic $\kkk$subgroup $\bf H$ of $\bf G$ such that ${\rm rank}_{\kkk}(\bf H)={\rm rank}_{\kkk}(\bf G)1$. We consider discrete subgroups $\Gamma$ of $G$ acting properly discontinuously on $G/H$ and we examine their images under a Cartan projection $\mu : G\rightarrow V^+$, where $V^+$ is a closed convex cone in a real finitedimensional vector space. We show that if $\Gamma$ is neither a torsion group nor a virtually cyclic group, then $\mu(\Gamma)$ is almost entirely contained in one connected component of $V^+\setminus C_H$, where $C_H$ denotes the convex hull of $\mu(H)$ in $V^+$. As an application, we describe all torsionfree discrete subgroups of $G\times G$ acting properly discontinuously on $G$ by left and right translation when ${\rm rank}_{\kkk}(\bf G)=1$. Keywords: Discrete subgroups of Lie groups, discrete subgroups of padic groups, reductive groups over local fields, properly discontinuous action, Cartan decomposition. MSC: 20G25, 22E40, 57S30 [ Fulltextpdf (243 KB)] for subscribers only. 