{
  "abstract": "We prove various properties of the universal valued Abelian groups $\\mathbb{G}_r(N)$ constructed by Niemiec, for $r\\in\\{1,\\infty\\}$ and $N\\in\\{0,2,3,\\ldots\\}$. First we show that the completion of $Γ_0=\\bigoplus_{n\\in\\mathbb{N}}\\mathbb{Q}/\\mathbb{Z}$ (for $N=0$) or $Γ_N=\\bigoplus_{n\\in\\mathbb{N}}\\mathbb{Z}/N\\mathbb{Z}$ (for $N\\geq 2$) with respect to a generic invariant metric, bounded by $1$ when $r=1$, is isometrically group-isomorphic to $\\mathbb{G}_r(N)$, recovering a result of Doucha in the case when $r=\\infty $ and $N=0$. This confirms an expectation of Doucha for the group $Γ_N$. We combine this genericity result with a criterion of Melleray and Tsankov concerning the extreme amenability of the generic completion of a countable group to obtain the extreme amenability of $\\mathbb{G}_r(N)$. We then establish the strictly stronger properties that these groups are Lévy and strongly exotic. We conclude by showing that $\\mathbb{G}_r(0)$ is also monothetic, from which we obtain the existence of a monothetic group structure on the Urysohn sphere, giving a bounded version of a result by Cameron and Vershik and answering a question of Niemiec.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25151",
  "author_total": 2,
  "authors": [
    {
      "name": "Alessandro Codenotti"
    },
    {
      "name": "Martino Lupini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45em67sh9vat8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25151",
  "fulltext_url": null,
  "id": "45em67sh9vat8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25151",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GN"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7214106302",
  "pdf_url": "https://arxiv.org/pdf/2609.25151",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45em67sh9vat8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25151",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25151"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The universal valued Abelian groups of Niemiec are Lévy, strongly exotic, extremely amenable, and $\\mathbb{G}_r(0)$ is generically monothetic",
  "title_text": "The universal valued Abelian groups of Niemiec are Lévy, strongly exotic, extremely amenable, and 𝔾_r(0) is generically monothetic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/45em67sh9vat8",
  "venue": "arXiv",
  "versions": []
}