Powers of commutators in infinite groups
Daan Heus
TL;DR
The paper classifies, for fixed integers $k,l,m\ge 2$ and $r$ coprime to them, when every group is $(k,l,m,r)$-quasi-Burnside or $(k,m,r)$-quasi-Honda, resolving the natural question about powers of commutators in infinite groups. It develops a framework linking these properties to universal triangle groups $B_{k,l,m}$ and $H_{k,m}$, and leverages the projective linear group $\mathrm{PSL}_2(\mathbb{R})$ to characterize when the Burnside-type conditions hold via the subgroup $M_G$ of units modulo $\mathrm{lcm}(k,l,m)$. A central technical contribution is the reduced-words apparatus and an elimination lemma that reduces solvability questions in free products to a finite set of equations in the base group, enabling precise equivalence results. The main findings show when all groups are quasi-Burnside or quasi-Honda and establish $\mathrm{PSL}_2(\mathbb{R})$ as a critical witness illustrating the necessity of the derived threshold and gcd conditions, with consequences for the structure of associated universal 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 $\mathrm{PSL}_2(\mathbb{R})$.