Every Lie group $G$ has this escape property. It says that  has the following property: for every $x \ne e$ in a sufficiently small neighborhood $U$ of the identity $e$ in $G$, there is a integer $n$ such that $x^n$ is not in $U$.
The question is that can we find sufficiently small neighborhood $V$ of $e$ in $G$, for any two different points $a,b \in V$ , there is a integer $m$ such that $a^m(b^{-1})^m$ is not in $V$.