A note on conciseness of Engel words

Gustavo A Fernandez-Alcober, Marta Moriagi, Gunnar Traustason

OAI: oai:purehost.bath.ac.uk:openaire_cris_publications/24c8b489-1b6a-44b4-9357-6a18e6320466 • DOI: https://doi.org/10.1080/00927872.2011.582061

It is still an open problem to determine whether the n-th Engel word [x,_n y] is concise, that is, if for every group G such that the set of values e_n(G) taken by [x,_n y] on G is finite it follows that the verbal subgroup E_n(G) generated by e_n(G) is also finite. We prove that if e_n(G) is finite then [En_(G),G] is finite, and either G=[E_n(G),G] is locally nilpotent and E_n(G) is finite, or G has a finitely generated section that is an infinite simple n-Engel group. It follows that [x_n y] is concise if n is at most four.