{
  "id": "1610.02079",
  "version": "v1",
  "published": "2016-10-06T21:44:54.000Z",
  "updated": "2016-10-06T21:44:54.000Z",
  "title": "Almost Engel compact groups",
  "authors": [
    "E. I. Khukhro",
    "P. Shumyatsky"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1512.06097",
  "categories": [
    "math.GR"
  ],
  "abstract": "We say that a group $G$ is almost Engel if for every $g\\in G$ there is a finite set ${\\mathscr E}(g)$ such that for every $x\\in G$ all sufficiently long commutators $[...[[x,g],g],\\dots ,g]$ belong to ${\\mathscr E}(g)$, that is, for every $x\\in G$ there is a positive integer $n(x,g)$ such that $[...[[x,g],g],\\dots ,g]\\in {\\mathscr E}(g)$ if $g$ is repeated at least $n(x,g)$ times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose ${\\mathscr E}(g)=\\{ 1\\}$ for all $g\\in G$.) We prove that if a compact (Hausdorff) group $G$ is almost Engel, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. If in addition there is a unform bound $|{\\mathscr E}(g)|\\leq m$ for the orders of the corresponding sets, then the subgroup $N$ can be chosen of order bounded in terms of $m$. The proofs use the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-06T21:44:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "engel compact groups",
      "engel groups",
      "locally nilpotent",
      "finite normal subgroup",
      "engel profinite groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}