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, (m) ...,G] is finite for some m ∈ N if and only if [w(G), G] is finite.The author is supported by the project PID2020-117281GB-I00 (Spanish Government, partially funded with ERDF) and by the research group 244 Álgebra. Aplicaciones (Public University of Navarra). Open Access funding provided by Universidad Publica de Navarra