Abstract
Every word in a free group F induces a probability measure on every finite group in a natural manner. It is an open problem whether two words that induce the same measure on every finite group, necessarily belong to the same orbit of AutF. A special case of this problem, when one of the words is the primitive word x, was settled positively by the third author and Parzanchevski [15]. Here we extend this result to the case where one of the words is xd or [x,y]d for an arbitrary d∈Z.
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