Given $\Gamma$ a discrete subgroup of a lie group $G$ I want to show that the action is wandering:
$\forall x\in G\exists U_x\vert \{\gamma\in\Gamma\vert \gamma U_x\cap U_x\neq \emptyset\}$ is finite
By considering a neighbourhood of the identity $V$ such that $V=V^{-1}=V^2$ and $V\cap \Gamma= e$ I have showed that every orbit $\Gamma y$ intersects $Vx$ in at most one element and then I got stuck any ulterior hints would be appreciated.