[2202.05044] On the Holosphere Density in Higher Order Homogeneous Space

Download the PDF of the paper by Or Landesberg and Hee Oh titled “On the Density of Holospheres in Higher Order Homogeneous Spaces”

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overview: Let $ G $ be the connected semisimple real algebraic group and $\Gamma < G$ the Zariski dense discrete subgroup. Let $N$ be the maximal astrological subgroup of $G$, and $P=MAN$ be the minimal parabolic subgroup, the normalizer of $N$. Let $\mathcal{E}$ represent the unique $P$ minimal subset of $\Gamma \backslash G$ and let $\mathcal{E}_0$ be the $P^\circ$ minimal subset . Consider the notion of limit points of the astrological sphere at the Fürstenberg boundary $ G/P $ and show that for any $ the following are equivalent:[g]\in \mathcal{E}_0$: (1) $gP\in G/P$ is the limit point of the astrology. (2) $[g]NM$ is clustered inside $\mathcal{E}$. (3) $[g]N$ is clustered at $\mathcal{E}_0$. The equivalence of (1) and (2) is due to Darbo for the rank 1 case. We also observe that unlike the convex cocompact group of rank-1 Lie groups, $NM$ minimality of $\mathcal{E}$ does not hold in general Anosov homogeneous spaces.