Powers with minimal commutator length in free products of groups

TL;DR

This paper determines the minimal commutator length of elements raised to a power in free products of groups, providing a comprehensive answer to a fundamental algebraic question.

Contribution

It offers a complete characterization of the minimal commutator length for elements in free products, extending understanding of algebraic properties in these groups.

Findings

Provides an explicit formula for minimal commutator length

Classifies elements based on their conjugacy properties

Enhances understanding of algebraic structure in free products

Abstract

Given a free product of groups $G = {\large *}_{j \in J} A_j$ and a natural number $n$, what is the minimal possible commutator length of an element $g^n \in G$ not conjugate to elements of the free factors? We give an exhaustive answer to this question.