On denseness of horospheres in higher rank homogeneous spaces

Let G be a connected semisimple real algebraic group and Γ < G be a Zariski dense discrete subgroup. Let N denote a maximal horospherical subgroup of G, and P = MAN the minimal parabolic subgroup which is the normalizer of N. Let E denote the unique P-minimal subset of Γ\G and let E₀ be a P^◦-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary G/P and show that the following are equivalent for any [g] ∈ E₀: (1) gP ∈ G/P is a horospherical limit point; (2) [g]NM is dense in E; (3) [g]N is dense in E₀. The equivalence of items (1) and (2) is due to Dal’bo in the rank one case. We also show that unlike convex cocompact groups of rank one Lie groups, the NM-minimality of E does not hold in a general Anosov homogeneous space.