Connection blocking in $$\text {SL}(n,\mathbb {R})$$ quotients

Abstract

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.