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- W2793711297 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 $B subset Msetminus {m_1, m_2 }$ 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 $ds^2=e^{-2z}dx^2+e^{2z}dy^2+dz^2$ includes copies of the hyperbolic plane, which makes studying its geometrical properties more interesting. In this paper we prove that all 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 $Sol/Gamma times Sol/Gamma$." @default.
- W2793711297 created "2018-03-29" @default.
- W2793711297 creator A5083799075 @default.
- W2793711297 date "2018-03-16" @default.
- W2793711297 modified "2023-09-27" @default.
- W2793711297 title "Connection Blocking In Quotients of $Sol$" @default.
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