{
  "abstract": "Let F be a free group. We present for arbitrary g\\in\\mathbb{N} a LOGSPACE (and thus polynomial time) algorithm that determines whether a given w\\in F is a product of at most g commutators; and more generally, an algorithm that determines, given w\\in F , the minimal g such that w may be written as a product of g commutators (and returns \\infty if no such g exists). This algorithm also returns words x_1,y_1,\\dots,x_g,y_g such that w=[x_1,y_1]\\dots[x_g,y_g] . These algorithms are also efficient in practice. Using them, we produce the first example of a word in the free group whose commutator length decreases under taking a square. This disproves in a very strong sense a conjecture by Bardakov.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "name": "Laurent Bartholdi"
    },
    {
      "name": "Sergei O. Ivanov"
    },
    {
      "name": "Danil Fialkovski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ct9k01k8gnhq.bib",
  "cited_by": 0,
  "doi": "10.4171/ggd/747",
  "fulltext_url": null,
  "id": "3ct9k01k8gnhq",
  "issns": [
    "1661-7207",
    "1661-7215"
  ],
  "landing_url": "https://hal.science/hal-05849733v1",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-gr"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W3206875207",
  "pdf_url": "https://ems.press/content/serial-article-files/28882",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": "European Mathematical Society",
  "ris": "https://research.draconitic.com/paper/3ct9k01k8gnhq.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:19Z",
      "native_id": "hal-05849733",
      "source": "hal",
      "url": "https://hal.science/hal-05849733v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-gr",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On commutator length in free groups",
  "title_text": "On commutator length in free groups",
  "type": "peer-reviewed",
  "updated_at": "2026-11-05T12:20:12Z",
  "url": "https://research.draconitic.com/paper/3ct9k01k8gnhq",
  "venue": "Groups, Geometry, and Dynamics",
  "versions": []
}