Let $H$ be a closed connected subgroup of the Lie group $G$ and $\Gamma$ be a lattice in $G$. Furthermore, assume that $H\cap \Gamma$ is a lattice in $H$. I wonder if it is true that the product $H\Gamma$ is a closed subset in $G$. (Or equivalently $H\Gamma/\Gamma$ is a closed subset in $G/\Gamma$).
Note that without the assumption that $H\cap \Gamma$ is a lattice in $H$ this is false since one can take $G=\mathbb R$, $H=<\pi>$ and $\Gamma=\mathbb Z$. (But note that $H\cap \Gamma=0$ in this case and thus not a lattice in $H$)