Powers of commutators in infinite groups

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 $k,m\ge 2$, 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 $k,l,m\ge 2$, and are given an integer $r$ that is coprime to $k,l$ and $m$, then there exist $x^{\mathrm{\prime }}$, $y^{\mathrm{\prime }}$ and $z^{\mathrm{\prime }}$ conjugate to respectively $x^r$, $y^r$ and $z^r$ such that $x^{\mathrm{\prime }}{y}^{\mathrm{\prime }}=z^{\mathrm{\prime }}$. 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 ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$.