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.