Research paper of a Lincoln algebraist in a high-ranking mathematical journal

New paper by Evgeny Khukhro (Univ. of Lincoln) and Pavel Shumyatsky (Univ. of Brasilia) “Strong conciseness of Engel words in profinite groups” has been accepted for publication in a high-ranking mathematical journal Mathematische Nachrichten. The work stems from the collaboration with University of Brasilia.

$Math-Nachr$ Abstract: A group word $w$ is said to be strongly concise in a class $\mathscr C$ of profinite groups if, for any group $G$ in $\mathscr C$ , either $w$ takes at least continuum many values in $G$ or the verbal subgroup $w(G)$ is finite. It is conjectured that all words are strongly concise in the class of all profinite groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture for multilinear commutator words, as well as for some other particular words. They also proved that every group word is strongly concise in the class of nilpotent profinite groups, as well as that 2-Engel words are strongly concise (but their approach does not seem to generalise to $n$ -Engel words for $n>2$ ). In the present paper we prove that for any $n$ the $n$ -Engel word $[...[x,y],y],\dots y]$ (where $y$ is repeated $n$ times) is strongly concise in the class of finitely generated profinite groups.