Resumen de Outer commutator words are uniformly concise

Gustavo Adolfo Fernández Alcober , Marta Morigi

• We prove that outer commutator words are uniformly concise, that is, if an outer commutator word ù takes m different values in a group G, then the order of the verbal subgroup ù(G) is bounded by a function depending only on m, and not on ù or G. This is obtained as a consequence of a structure theorem for the subgroup ù(G), which is valid if G is soluble, and without assuming that ù takes finitely many values in G. More precisely, there is an abelian series of ù(G), such that every section of the series can be generated by values of ù all of whose powers are also values of ù in that section. For the proof of this latter result, we introduce a new representation of outer commutator words by means of binary trees, and we use the structure of the trees to set up an appropriate induction.