{
  "abstract": "We introduce the notion of a Magnus marking for a finite generating set of a group and prove a certain expansion property. Using this property, we determine the second and third smallest growth rates of the rank-$d$ free group for $d\\ge 2$. We also give a new lower bound for the smallest growth rate of the genus-$g$ surface group for $g\\ge 2$, as well as a lower bound for the growth rate associated with a one-relator presentation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.17444",
  "author_total": 3,
  "authors": [
    {
      "name": "Koji Fujiwara"
    },
    {
      "name": "Sang-hyun Kim"
    },
    {
      "name": "Ryokichi Tanaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0d0fb9tp4rw88.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.17444",
  "fulltext_url": null,
  "id": "0d0fb9tp4rw88",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.17444",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7169835943",
  "pdf_url": "https://arxiv.org/pdf/2607.17444",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0d0fb9tp4rw88.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2607.17444",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.17444"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Small growth rates of free groups",
  "title_text": "Small growth rates of free groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/0d0fb9tp4rw88",
  "venue": "arXiv",
  "versions": []
}