On denseness of horospheres in higher rank homogeneous spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866914650005700608 |
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| author | Landesberg, Or Oh, Hee |
| author_facet | Landesberg, Or Oh, Hee |
| contents | 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 $\mathcal{E}$ denote the unique $P$-minimal subset of $Γ\backslash G$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_05044 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On denseness of horospheres in higher rank homogeneous spaces Landesberg, Or Oh, Hee Dynamical Systems Geometric Topology 37A17 (primary) 22F30, 22E40 (secondary) 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 $\mathcal{E}$ denote the unique $P$-minimal subset of $Γ\backslash G$ 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. |
| title | On denseness of horospheres in higher rank homogeneous spaces |
| topic | Dynamical Systems Geometric Topology 37A17 (primary) 22F30, 22E40 (secondary) |
| url | https://arxiv.org/abs/2202.05044 |