Groups with finitely many long commutators of maximal order
Iker de las Heras, Federico Di Concilio, Pavel Shumyatsky
TL;DR
The paper extends the study of groups with finitely many commutators of maximal order to commutators of length \\ell\\ (with \\ell\\ge 3). It develops a framework around the finite set \\mathcal{D}_{\\ell} of maximal-order length-\\\\ell\\-commutators and analyzes the associated subgroup D. Under residually finite (and finitely generated) hypotheses, it shows the existence of a finite-index subgroup M with \\gamma_{\\ell}(M)=1 and, in many cases, a finite, (m,\\ell,r)-bounded order for \\gamma_{\\ell}(G). A key contribution is the introduction of d-stability and its use to obtain bounded-index subgroups with controlled lower central series under explicit arithmetic and nilpotence conditions, yielding concrete finiteness and boundedness results for the relevant quotients.
Abstract
Given a group $G$ and elements $x_1,x_2,\dots, x_\ell\in G$, the commutator of the form $[x_1,x_2,\dots, x_\ell]$ is called a commutator of length $\ell$. The present paper deals with groups having only finitely many commutators of length $\ell$ of maximal order. We establish the following results. Let $G$ be a residually finite group with finitely many commutators of length $\ell$ of maximal order. Then $G$ contains a subgroup $M$ of finite index such that $γ_\ell(M)=1$. Moreover, if $G$ is finitely generated, then $γ_\ell(G)$ is finite. Let $\ell,m,n,r$ be positive integers and $G$ an $r$-generator group with at most $m$ commutators of length $\ell$ of maximal order $n$. Suppose that either $n$ is a prime power, or $n=p^αq^β$, where $p$ and $q$ are odd primes, or $G$ is nilpotent. Then $γ_\ell(G)$ is finite of $(m,\ell,r)$-bounded order and there is a subgroup $M\le G$ of $(m,\ell,r)$-bounded index such that $γ_\ell(M)=1$.