Powers of commutators in infinite groups

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 $\mathrm{PSL}_2(\mathbb{R})$.