On generalisations of conciseness

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 $$1/m$$ 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, ⁽ᵐ⁾, G] [ w ( G ) , G , ⋯ ( m ) , G ] is finite for some m ∈ N m ∈ N if and only if $$[w(G), G]$$ [ w ( G ) , G ] is finite.