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 $$1/m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> -concise word is semiconcise by proving that when the group-word w takes finitely many values in G , the iterated commutator subgroup $$[w(G), G, {\mathop {\dots }\limits ^{(m)}}, G]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>w</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mo>⋯</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> is finite for some $$m \in \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> if and only if $$[w(G), G]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>w</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> is finite.