School of Mathematics & Physics, University of Lincoln
The paper by Evgeny Khukhro and Pavel Shumyatsky “Almost Engel compact groups”, http://dx.doi.org/10.1016/j.jalgebra.2017.04.021, has just been accepted for publication in Journal of Algebra.
(See also https://arxiv.org/pdf/1610.02079.pdf.)
Abstract: We say that a group is almost Engel if for every there is a finite set such that for every all sufficiently long commutators belong to , that is, for every there is a positive integer such that if is repeated at least times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose for all $g\in G$.) We prove that if a compact (Hausdorff) group is almost Engel, then has a finite normal subgroup such that is locally nilpotent. If in addition there is a unform bound for the orders of the corresponding sets, then the subgroup can be chosen of order bounded in terms of . The proofs use the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent.