{
  "abstract": "We prove that for every non-elementary torsion-free hyperbolic group $H$ and sufficiently large odd $n>1$, the quotient $H/H^n$ has cost $1$. In particular, when $H$ is the free group of rank $m\\ge 2$, we obtain that the free Burnside group $B(m,n)$ has cost $1$ for every odd exponent $n\\geq 665$. It follows that $B(m,n)$ is anti-treeable, and that its first $\\ell^2$-Betti number $β_1^{(2)}(B(m,n))$ vanishes. The latter recovers and extends a result of Feldkamp and Kionke [Proc. Amer. Math. Soc., 2023], who proved that $β_1^{(2)}(B(m,p))=0$ for all sufficiently large prime exponents $p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.20472",
  "author_total": 2,
  "authors": [
    {
      "name": "Miguel Donoso-Echenique"
    },
    {
      "name": "Eduardo Silva"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fb3x396cwxm2v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.20472",
  "fulltext_url": null,
  "id": "fb3x396cwxm2v",
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  "landing_url": "https://arxiv.org/abs/2608.20472",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.PR"
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    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
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  "openalex": "W7204111129",
  "pdf_url": "https://arxiv.org/pdf/2608.20472",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fb3x396cwxm2v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2608.20472",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.20472"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Free Burnside groups of large odd exponent have cost 1",
  "title_text": "Free Burnside groups of large odd exponent have cost 1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/fb3x396cwxm2v",
  "venue": "arXiv",
  "versions": []
}