论文标题
关于高级halosphers的密度
On denseness of horospheres in higher rank homogeneous spaces
论文作者
论文摘要
令$ g $为连接的半胶实代数组，$γ<g $为zariski密集的离散子组。令$ n $表示$ g $的最大halosphical子组，$ p = man $最小抛物线子组，是$ n $的标准化器。令$ \ Mathcal {e} $表示$γ\ Backslash G $的唯一$ P $ - 米数子集，让$ \ Mathcal {e} _0 $为$ p^\ circ $ -minimal subset。我们考虑了furstenberg边界中的h鼠极限点$ g/p $的概念，并表明以下含量是等于\ Mathcal {e} {e} _0 $：（1）$ gp \ in g/p $中的任何$ [g] \ in g/p $中的等效物； （2）$ [g] nm $在$ \ mathcal {e} $中密集； （3）$ [g] n $在$ \ MATHCAL {E} _0 $中密集。 （1）和（2）的等效性是由于排名中的Dal'bo引起的。我们还观察到，与凸的cooCompact组排名一组不同，$ nm $ $ \ Mathcal {e} $的$ nm $ - 最小程度不在一般的Anosov同质空间中。
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.
