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.