Every Lie group $G$ has the following escape 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 if we can find a 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 $V$.

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If G is Commutative group， the question actually is true, since G has escape property. I guess it is true for some class except Commutative group.  The problem is to  state that  Power mapping can enlarge the distance between two different points.
But,I didn't find a good property of power mapping by looking up data.  meanwhile，Thanks  Michael Albanese for editing of question.