Quasiperiodic and mixed commutator factorizations in free products of groups

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.