Powers of commutators in infinite groups

TL;DR

This paper investigates the behavior of powers of commutators and products in finite groups, providing conditions under which certain conjugacy and product properties hold, using combinatorial group theory and properties of PSL(2,R).

Contribution

It offers a complete characterization of when groups exhibit these commutator and product power properties based on parameters k, l, m, r.

Findings

Identifies conditions on k, l, m, r for properties to hold in all groups.

Uses combinatorial group theory and PSL(2,R) to prove results.

Provides a comprehensive answer to the natural question about powers of commutators and products.

Abstract

Given elements $x,u,z$ in a finite group $G$ such that $z$ is the commutator of $x$ and $u$, and the orders of $x$ and $z$ divide respectively integers $k,m \geq 2$, and given an integer $r$ that is coprime to $k$ and $m$, there exists $w \in G$ such that the commutator of $x^r$ and $w$ is conjugate to $z^r$. If instead we are given elements $x,y,z \in G$ such that $xy = z$, whose respective orders divide integers $k,l,m \geq 2$, and are given an integer $r$ that is coprime to $k,l$ and $m$, then there exist $x'$, $y'$ and $z'$ conjugate to respectively $x^r$, $y^r$ and $z^r$ such that $x'y' = z'$. In this paper we completely answer the natural question for which values of $k,l,m,r$ every group has these properties. The proof uses combinatorial group theory and properties of the projective special linear group $\mathrm{PSL}_2(\mathbb{R})$.