Divergent orbits on S-adic homogeneous spaces

Let G be a semisimple algebraic group defined over a number field K and let S be a finite set of non-equivalent valuations of K containing the archimedean ones. Set G = Pi(upsilon epsilon S) G(K-upsilon) and Gamma = G(O) where O is the ring of S-integers of K. Fix upsilon epsilon S and a K-upsilon-split algebraic torus T-upsilon of G(K-upsilon). In this paper, in complement to results from [To], we prove results about the divergent orbits for the action of T-upsilon on G/Gamma by left translation.