On denseness of horospheres in higher rank homogeneous spaces

• Published in: Ergodic theory and dynamical systems Vol. 44; no. 11; pp. 3272 - 3289
• Main Authors: LANDESBERG, OR, OH, HEE
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.11.2024
• Subjects: Original Article; horospheres; limit cones; horospherical limit points; 22E40; infinite volume; 37A17; 22F30
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2024.12

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.