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

## TL;DR

This paper investigates whether words in free groups are uniquely determined by the probability measures they induce on all finite groups, extending previous results to specific classes of words.

## Contribution

It extends prior work by proving that words of the form x^d or [x,y]^d are uniquely determined by their induced measures on finite groups.

## Key findings

- Words x^d and [x,y]^d are determined by their measures on finite groups.
- Generalizes previous results for primitive words to specific non-primitive words.
- Provides partial progress on the open problem regarding word orbits and induced measures.

## 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 $d\in\mathbb{Z}$.