On denseness of horospheres in higher rank homogeneous spaces

TL;DR

This paper explores the denseness properties of horospheres in higher rank homogeneous spaces, establishing equivalences between boundary limit points and the density of certain orbits in the space.

Contribution

It generalizes known rank one results to higher rank spaces, characterizing horospherical limit points via orbit density in Anosov homogeneous spaces.

Findings

Equivalence between horospherical limit points and orbit density in the space.

Extension of Dal'bo's rank one results to higher rank homogeneous spaces.

Observation that $NM$-minimality does not generally hold in higher rank Anosov 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 (1) and (2) is due to Dal'bo in the rank one case. We also observe 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.