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 $[x_1,y_1]\dots [x_k,y_k]=z^n$, 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.

14 pages, 5 figures. V3: minor corrections

References in corpus (2)

• How to generalize known results on equations over groups
• The SQ-universality of one-relator relative presentations

Cited by in corpus (3)

• Scl in graphs of groups
• Commutator length of powers in free products of groups
• Powers with minimal commutator length in free products of groups