The words x^d and [x,y]^d are rigid: any word inducing the same measures on all finite groups is an automorphic image of them.
Torsion elements and centralizers in free products of profinite groups
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Some Orbits of Free Words that are Determined by Measures on Finite Groups
The words x^d and [x,y]^d are rigid: any word inducing the same measures on all finite groups is an automorphic image of them.