Divergent Orbits on S-adic Homogeneous Spaces

Abstract

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 = ∏ v∈S G(Kv) and Γ = G(O) where O is the ring of S-integers of K. Fix v ∈ S and a Kv-split algebraic torus Tv of G(Kv). In this paper, in complement to results from [To], we prove results about the divergent orbits for the action of Tv on G/Γ by left translation.