On generalized concise words

The study of verbal subgroups within a group is well known for being an effective tool to obtain structural information about a group. Therefore, conditions that allow the classification of words in a free group are of paramount importance. One of the most studied problems is to establish which words are concise, where a word 𝑤 is said to be concise if the verbal subgroup w ⁢ ( G ) w(G) is finite in each group 𝐺 in which 𝑤 takes only a finite number of values. The purpose of this article is to present some results, in which a hierarchy among words is introduced, generalizing the concept of concise word.