On generalisations of conciseness

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.