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Journal of Lie Theory 18 (2008), No. 4, 961--978
Copyright Heldermann Verlag 2008

Proper Actions on Corank-One Reductive Homogeneous Spaces

Fanny Kassel
Dép. de Mathématiques, Bâtiment 425, Faculté des Sciences, Université Paris-Sud 11, 91405 Orsay, France


\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 finite-dimensional 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 torsion-free 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 p-adic groups, reductive groups over local fields, properly discontinuous action, Cartan decomposition.

MSC: 20G25, 22E40, 57S30

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