# Quasiperiodic and mixed commutator factorizations in free products of groups

## TL;DR

This paper generalizes the known fact that nontrivial commutators in free groups are not proper powers, providing new results on factorizations and equations involving commutators in free products of groups.

## Contribution

It introduces a theorem extending the non-power property of commutators to free products and explores implications for equations and factorizations involving commutators.

## Key findings

- A commutator in a free product cannot be a proper power under certain conditions.
- Solutions to specific equations imply elements are conjugate to free factors.
- Commutator factorizations into conjugate elements have distinct factors.

## Abstract

It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation $[ x_1, y_1] \ldots [ x_k, y_k] = z^n$, where $n \ge 2k$, in a free product $\mathcal{F}$ of groups without nontrivial elements of order $\le n$ implies that $z$ is conjugate to an element of a free factor of $\mathcal{F}$. If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct.