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Michael Albanese
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About Lie group G has this escape property?

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

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