On generalisations of conciseness

TL;DR

This paper explores the properties of conciseness and semiconciseness in group words, demonstrating their differences and establishing conditions under which certain words are semiconcise or concise.

Contribution

It proves that conciseness and semiconciseness are not equivalent and characterizes when a word is semiconcise based on finiteness conditions in group theory.

Findings

A specific word by Ol'shanskii is semiconcise but not concise.

Every 1/m-concise word is semiconcise.

Finiteness of certain iterated commutator subgroups characterizes semiconciseness.

Abstract

Based on the notions of conciseness and semiconciseness, we show that these properties are not equivalent by proving that a word originally presented by Ol'shanskii is semiconcise but not concise. We further establish that every $1/m$-concise word is semiconcise by proving that when the group word $w$ takes finitely many values in $G$, the iterated commutator subgroup $[w(G), G, \stackrel{(m)}{\dots}, G]$ is finite for some $m \in \mathbb{N}$ if and only if $[w(G), G]$ is finite.