20.89 (2022)

An element $g$ of a group $G$ is said to be almost Engel if there is a finite subset $\mathcal{E}(g)$ of $G$ such that for every $x \in G$ all sufficiently long commutators $[\dots[[x, g], g], \dots, g]$ belong to $\mathcal{E}(g)$, that is, there is a positive integer $n(x, g)$ such that $[\dots[[x, g], g], \dots, g] \in \mathcal{E}(g)$ if $g$ is repeated $\geqslant n(x, g)$ times. An element $g$ is Engel if we can take $\mathcal{E}(g) = \{1\}$. The set of Engel elements of a linear group is a subgroup by a well-known result of Gruenberg (J. Algebra, 3 (1966), 291–303). Is the set of almost Engel elements of a linear group a subgroup?

A linear group in which all elements are almost Engel is finite-by-hypercentral (P. Shumyatsky, Monatsh. Math., 186 (2018), 711–719).