On rational and concise words

TL;DR

This paper explores the concept of concise and rational words in group theory, establishing conditions under which words are concise in residually finite groups and introducing new concise words.

Contribution

It demonstrates that rational words are concise in residually finite groups and provides a new sufficient condition for a word to be rational, leading to a new family of concise words.

Findings

Rational words are concise in residually finite groups.

A new sufficient condition for a word's rationality is established.

A new family of concise words in residually finite groups is introduced.

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. It is still an open problem whether every word is concise in the class of residually finite groups. A word w is rational if the number of solutions to the equation w=g is the same as the number of solutions to w=g^e for every finite group G and for every e relatively prime to |G|. We observe that any rational word is concise in the class of residually finite groups. Further we give a sufficient condition for rationality of a word. This is used to produce a new family of words that are concise in the class of residually finite groups.