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Let $G$ be a real connected Lie group with a left invariant metric $d$, $\mathfrak{g}$ its Lie algebra. In this paper we present a set of interesting upper and lower bounds for $|d\exp_{x}(y)|,\ x,y \in \mathfrak{g}$. If $\textrm{ad}_x$ is diagonalizable, these bounds only depend on eigenvalues of $\textrm{ad}_x$, but in general they are functions of the singular values $\textrm{ad}_x$., Comment: 7 pages

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 $ds^2=e^{-2z}dx^2+e^{2z}dy^2+dz^2$ 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 $Sol/\Gamma \times Sol/\Gamma$., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1706.07996; text overlap with arXiv:1211.7291 by other authors

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., Comment: 16 pages