Commutator length of powers in free products of groups

TL;DR

This paper investigates the minimal commutator length of powers of elements in free products of groups, providing near-complete answers and exploring related problems in group theory.

Contribution

It offers a nearly complete characterization of the minimal commutator length for powers in free products, advancing understanding of conjugacy and commutator properties in such groups.

Findings

The minimal commutator length is one of two possible values depending on the factors.

Provides an almost complete answer to a specific open problem.

Explores related problems in the structure of free product groups.

Abstract

Given groups $A$ and $B$, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$). Other similar problems are also considered.