{
  "abstract": "Let $G$ be a random group in the few-relator model with $n\\ge 2$ generators and $m\\ge 1$ relators. We prove that asymptotically almost surely the first $\\ell^2$-Betti number, the cost minus one, and the absolute rank gradient of $G$ are all equal to $\\max\\{n-m-1,0\\}$. In particular, the equality $β_1^{(2)}(G)=0$ when $m\\ge n-1$ answers positively a conjecture of Kielak, Kropholler and Wilkes [J. Lond. Math. Soc., 2022].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29502",
  "author_total": 2,
  "authors": [
    {
      "name": "Miguel Donoso-Echenique"
    },
    {
      "name": "Eduardo Silva"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5z7qb967x3r7h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29502",
  "fulltext_url": null,
  "id": "5z7qb967x3r7h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29502",
  "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"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7223216367",
  "pdf_url": "https://arxiv.org/pdf/2610.29502",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5z7qb967x3r7h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29502",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29502"
    }
  ],
  "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": "The first $\\ell^2$-Betti number, cost and rank gradient of few-relator random groups",
  "title_text": "The first ℓ^2-Betti number, cost and rank gradient of few-relator random groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/5z7qb967x3r7h",
  "venue": "arXiv",
  "versions": []
}