{
  "id": "2111.15277",
  "version": "v2",
  "published": "2021-11-30T10:51:39.000Z",
  "updated": "2022-01-19T11:22:51.000Z",
  "title": "Markov's problem for free groups",
  "authors": [
    "Dmitri Shakhmatov",
    "Víctor Hugo Yañez"
  ],
  "comment": "Sections 2 and 11 have been added",
  "categories": [
    "math.GR",
    "math.AG",
    "math.CO",
    "math.GN",
    "math.LO"
  ],
  "abstract": "We prove that every unconditionally closed subset of a free group is algebraic, thereby answering affirmatively a 76 years old problem of Markov for free groups. In modern terminology, this means that Markov and Zariski topologies coincide in free groups. It follows that the class of groups for which Markov and Zariski topologies coincide is not closed under taking quotients. We also show that Markov and Zariski topologies differ from the so-called precompact Markov topology in non-commutative free groups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-19T11:22:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E05",
      "03C55",
      "03E05",
      "03E75",
      "14L99",
      "20E06",
      "22A05",
      "54E50",
      "54H11"
    ],
    "keywords": [
      "markovs problem",
      "zariski topologies coincide",
      "precompact markov topology",
      "zariski topologies differ",
      "old problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}