{
  "abstract": "We prove that almost sure topological finite generation holds for a general class of random models of profinite groups. We deduce that in the models introduced by Liu--Wood and Sawin--Wood, one has that finite presentation holds almost surely, with almost sure control of the deficiency in the presentation. In particular this settles questions raised in work of Liu--Wood and Sawin--Wood. Using the same underlying principle, we show that there exists a unique universal d-generated perfect profinite group: its finite quotients are precisely the finite d-generated perfect groups. This settles a conjecture of Nikolov. We show in addition that this group is projective and admits a profinite presentation with $d$ generators and $d$ relations, and with no fewer relations on $d$ generators. The main underlying theme is to bring in an insight from crown theory: groups with high rank are always witnessed by a crown with large multiplicity. This approach was discovered independently by ChatGPT5.5 pro and Aletheia, an internal agent at Google DeepMind.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30200",
  "author_total": 2,
  "authors": [
    {
      "name": "Carlo Pagano"
    },
    {
      "name": "Mark Shusterman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0w7d8arj830q0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30200",
  "fulltext_url": null,
  "id": "0w7d8arj830q0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30200",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214386682",
  "pdf_url": "https://arxiv.org/pdf/2609.30200",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0w7d8arj830q0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.30200",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30200"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "High Rank and Multiplicity in Random and Perfect Profinite Groups",
  "title_text": "High Rank and Multiplicity in Random and Perfect Profinite Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/0w7d8arj830q0",
  "venue": "arXiv",
  "versions": []
}