{
  "abstract": "A group of conformal automorphisms of a Riemann surface is said to act purely non-free, or to produce a gpnf-action, if each of its finite-order elements has fixed points. It is known that every finite group admits a gpnf-action on a suitable closed Riemann surface of genus at least two. If $G$ is an infinite countable group, and $X$ is an infinite-type orientable surface, without planar ends and whose space of ends is self-similar, then we observe that there is a Riemann surface structure on $X$ admitting a gpnf-action by $G$. We obtain upper bounds for those finite groups producing a gpnf-action on a closed Riemann surface of genus $g \\geq 2$. If $G$ is a finite group, then its purely non-free genus $σ^{pnf}_{hyp}(G)$ is the minimum genus of at least two of a closed Riemann surface admitting a gpnf-action by $G$. This parameter is already known for cyclic, abelian non-cyclic groups, ${\\mathcal A}_{4}$, ${\\mathcal A}_{5}$ and ${\\mathfrak S}_{4}$. For general $G$, we provide the value of $σ^{pnf}_{hyp}(G)$, correcting a previous statement on this value. In particular, we observe that every genus is the purely non-free genus of some group. As an application, we describe the minimal purely non-free actions for the family of metacyclic groups $G^{0}_{s}(n)=\\langle x,y: x^{n}=y^{2}=1, yxy=x^{s}\\rangle,$ where $n \\geq 3$ and $2 \\leq s \\leq n-1$ is such that $s^{2} \\equiv 1 \\mod(n)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17405",
  "author_total": 2,
  "authors": [
    {
      "name": "Rubén A. Hidalgo"
    },
    {
      "name": "Jhony S. Leyca"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/51wq278ybmkza.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17405",
  "fulltext_url": null,
  "id": "51wq278ybmkza",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17405",
  "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.CV"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7221822729",
  "pdf_url": "https://arxiv.org/pdf/2610.17405",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/51wq278ybmkza.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17405",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17405"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Purely non-free actions on Riemann surfaces",
  "title_text": "Purely non-free actions on Riemann surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/51wq278ybmkza",
  "venue": "arXiv",
  "versions": []
}