Some Orbits of Free Words that are Determined by Measures on Finite Groups

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 $\mathrm{Aut}F$. 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 [arXiv:1202.3269]. Here we extend this result to the case where one of the words is $x^d$ or ${\left[x,y\right]}^d$ for an arbitrary $d\in \mathbb{Z}$.

arXiv comment16 pages, minor changes, journal version

References in corpus (8)

• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• Measure Preserving Words are Primitive
• Primitive Words, Free Factors and Measure Preservation
• Words Maps and Spectra of Random Graph Lifts
• Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$
• Word Measures on Symmetric Groups
• Surface Words are Determined by Word Measures on Groups
• Word Measures on Unitary Groups

Cited by in corpus (5)

• Word Measures on Symmetric Groups
• Surface Words are Determined by Word Measures on Groups
• Word Measures on $G{L}_N(q)$ and Free Group Algebras
• Automorphism-invariant positive definite functions on free groups
• A note on words having the same image on finite groups