  Let k be a local field and G the set of k-points of a connected semisimple
algebraic k-group of rank one. We describe all torsion-free discrete subgroups
of G\times G acting properly discontinuously on G by left and right
multiplication. To this end, we prove a general result on the Cartan projection
of discrete groups acting properly discontinuously on corank-one reductive
homogeneous spaces over k.
