On finiteness of verbal subgroups

TL;DR

This paper investigates the finiteness properties of verbal subgroups in groups, focusing on the conjecture that certain non-commutator words produce concise verbal subgroups.

Contribution

It provides new results supporting the conjecture that specific non-commutator words are concise in all groups.

Findings

Support for the conjecture on non-commutator words

Several new theorems on the finiteness of verbal subgroups

Advances towards understanding verbal subgroup properties

Abstract

Given a group-word $w$ and a group $G$, the set of $w$-values in $G$ is denoted by $G_w$ and the verbal subgroup $w(G)$ is the one generated by $G_w$. The word $w$ is concise if $w(G)$ is finite for all groups $G$ in which $G_w$ is finite. We obtain several results supporting the conjecture that the word $[u_1,\dots,u_s]$ is concise whenever the words $u_1,\dots,u_s$ are non-commutator.