Connection blocking in quotients of Sol

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 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, otherwise we call it non-blockable.

Sol is an important Lie group and one of the eight homogeneous Thurston 3-geometries. It is a unimodular solvable Lie group diffeomorphic to $R^3$, and together with the left invariant metric $d{s}^2=e^{-2z}d{x}^2+e^{2z}d{y}^2+d{z}^2$ includes copies of the hyperbolic plane, which makes studying its geometrical properties more interesting. In this paper we prove that all lattice quotients of Sol are non-blockable. In particular, we show that for any lattice $\Gamma \subset Sol$, the set of non-blockable pairs is a dense subset of $Sol/\Gamma \times Sol/\Gamma$.