Proper Actions on Corank-One Reductive Homogeneous Spaces

Let k be a local field, G the set of k-points of a connected semisimple algebraic k-group G , and H the set of k-points of a connected reductive algebraic k-subgroup H of G such that rank k (H) = rank k (G) -1 .We consider discrete subgroups of G acting properly discontinuously on G/H and we examine their images under a Cartan projection : G V + , where V + is a closed convex cone in a real finite-dimensional vector space.We show that if is neither a torsion group nor a virtually cyclic group, then () is almost entirely contained in one connected component of V + \ C H , where C H denotes the convex hull of (H) in V + .As an application, we describe all torsion-free discrete subgroups of G G acting properly discontinuously on G by left and right translation when rank k (G) = 1 .