On words that are concise in residually finite groups

TL;DR

This paper investigates the conciseness of certain group words in residually finite groups, proving that specific multilinear commutator words raised to prime-power powers are concise or boundedly concise.

Contribution

It demonstrates that for multilinear commutator words w and prime powers q, the word w^q is concise in residually finite groups, advancing understanding of word properties in group theory.

Findings

w^q is concise for multilinear commutator w and prime-power q in residually finite groups

w^q is boundedly concise when w is a lower central word γ_k

It remains open whether w^q is concise in all groups

Abstract

A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group $G\in X$, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime-power, then the word $w^q$ is indeed concise in the class of residually finite groups. Further, we show that in the case where $w=\gamma_{k}$ the word $w^q$ is boundedly concise in the class of residually finite groups. It remains unknown whether the word $w^q$ is actually concise in the class of all groups.