{
  "id": "2308.02209",
  "version": "v1",
  "published": "2023-08-04T08:51:18.000Z",
  "updated": "2023-08-04T08:51:18.000Z",
  "title": "Varieties of groups and the problem on conciseness of words",
  "authors": [
    "Cristina Acciarri",
    "Pavel Shumyatsky"
  ],
  "comment": "22 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "A group-word $w$ is concise in a class of groups $\\mathcal X$ if and only if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G\\in \\mathcal X$. It is a long-standing open problem whether every word is concise in residually finite groups. In this paper we observe that the conciseness of a word $w$ in residually finite groups is equivalent to that in the class of virtually pro-$p$ groups. This is used to show that if $q,n$ are positive integers and $w$ is a multilinear commutator word, then the words $w^q$ and $[w^q,_{n} y]$ are concise in residually finite groups. Earlier this was known only in the case where $q$ is a prime power. In the course of the proof we establish that certain classes of groups satisfying the law $w^q\\equiv1$, or $[\\delta_k^q,{}_n\\, y]\\equiv1$, are varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-04T08:51:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E26",
      "20F10",
      "20F40",
      "20E10",
      "20F45"
    ],
    "keywords": [
      "residually finite groups",
      "conciseness",
      "multilinear commutator word",
      "prime power",
      "verbal subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}