On denseness of horospheres in higher rank homogeneous spaces

Abstract

Let $ G $ be a connected semisimple real algebraic group and $\Gamma <G$ 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 $\mathcal E$ denote the unique P-minimal subset of $\Gamma \backslash G$ and let $\mathcal E_0$ be a $P^\circ $ -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 $[g]\in \mathcal E_0$ : (1) $gP\in G/P$ is a horospherical limit point; (2) $[g]NM$ is dense in $\mathcal E$ ; (3) $[g]N$ is dense in $\mathcal E_0$ . 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 $\mathcal E$ does not hold in a general Anosov homogeneous space.