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About Lie group G has this escape property?

Every Lie group G has this escape property. It says that has the following property: for every x ≠ 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.

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