Connection Blocking in $SL(n,R)$ Quotients

TL;DR

This paper studies the blocking properties of connection curves in homogeneous spaces formed by quotients of $SL(n,R)$, showing that while some pairs are densely blockable in $SL(2,R)/ ext{lattice}$, the spaces are generally not finitely blockable.

Contribution

It characterizes blocking properties of $SL(n,R)/ ext{lattice}$ spaces, proving non-blockability for higher dimensions and analyzing quaternionic structures for co-compact lattices.

Findings

Set of blockable pairs is dense in $M_2$

Manifolds $M_n$ are not blockable

Quotients with quaternionic structures are not finitely blockable

Abstract

Let $G$ be a connected Lie group and $\Gamma \subset G$ a lattice. Connection curves of the homogeneous space $M=G/\Gamma$ are the orbits of one parameter subgroups of $G$. To \textit{block} a pair of points $m_1,m_2 \in M$ is to find a \textit{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 \textit{blockable} if every pair of points in $M$ can be blocked. \par In this paper we investigate blocking properties of $M_n=SL(n,R)/\Gamma$, where $\Gamma=SL(n,Z)$ is the integer lattice. We focus on $M_2$ and show that the set of bloackable 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 $SL(2,R)$ and a way for making co-compact lattices in this context. We show that the obtained quotient homogeneous spaces are not finitely blockable.