Let G be a connected Lie group and $$\varGamma \subset G$$ a lattice. Connection curves of the homogeneous space $$M=G/\varGamma $$ are the orbits of one parameter subgroups of G. To block a pair of points $$m_1,m_2 \in M$$ is to find a finite set $$B \subset M{\setminus } \{m_1, m_2 \}$$ such that every connecting curve joining $$m_1$$ and $$m_2$$ intersects B. The homogeneous space M is blockable if every pair of points in M can be blocked. In this paper we investigate blocking properties of $$M_n= \text {SL}(n,\mathbb {R})/\varGamma $$, where $$\varGamma =\text {SL}(n,\mathbb {Z})$$ is the integer lattice. We focus on $$M_2$$ and show that the set of non blackable pairs is a dense subset of $$M_2 \times M_2$$, and we conclude manifolds $$M_n$$ are not blockable. Finally, we review a quaternionic structure of $$\text {SL}(2,\mathbb {R})$$ and a way for making co-compact lattices in this context. We show that the obtained quotient homogeneous spaces are not finitely blockable.

中文翻译：

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$$\text {SL}(n,\mathbb {R})$$SL(n,R) 商数中的连接阻塞

令 G 为连通李群，$$\varGamma \subset G$$ 为格。齐次空间的连接曲线 $$M=G/\varGamma $$ 是 G 的一个参数子群的轨道。 阻塞一对点 $$m_1,m_2 \in M$$ 是找到一个有限集 $$ B \subset M{\setminus } \{m_1, m_2 \}$$ 使得每条连接 $$m_1$$ 和 $$m_2$$ 的连接曲线都与 B 相交。如果 M 中的每对点都可以阻塞，则齐次空间 M 是可阻塞的可以被阻止。在本文中，我们研究了 $$M_n= \text {SL}(n,\mathbb {R})/\varGamma $$ 的阻塞特性，其中 $$\varGamma =\text {SL}(n,\mathbb {Z })$$ 是整数格。我们专注于 $$M_2$$ 并表明不可黑化对的集合是 $$M_2 \times M_2$$ 的密集子集，我们得出的结论是流形 $$M_n$$ 是不可阻塞的。最后，我们回顾了 $$\text {SL}(2, \mathbb {R})$$ 以及在这种情况下制作协同紧凑格的方法。我们证明所获得的商齐次空间不是有限可分的。