On finitely generated Engel branch groups (Q6618668)

An element \(h\) of a group \(G\) is said to be a \textit{left Engel element} if there is a number \(n\) such that the \(n\) times iterated commutator \([\ldots[[g,h], h]\ldots]\) is trivial for all \(g\in G\). The group \(G\) is called an \textit{Engel-group} if every element is a left Engel element.\N\NIn the present work, the author constructs examples of branch groups that are also Engel groups. These examples are finitely generated \(p\)-groups and provide an answer to Questions 2 and 3 in [\textit{G. A. Fernandés-Alcober} et al., J. Algebra 554, 54--77 (2020; Zbl 1452.20034)]. These examples are not nilpotent and the only other known examples on non-nilpotent Engel groups are the examples constructed by \textit{E. S. Golod} [Transl., Ser. 2, Am. Math. Soc. 84, 83--88 (1968; Zbl 0206.32402); translation from Tr. Mezdunarod. Kongr. Mat., Moskva 1966, 284--289 (1968)].