{
  "abstract": "For every $n \\geq 1$ there is a distinguished $\\mathfrak{sl}_2(\\mathbb{R})$ action on the Poincaré disk, arising as the infinitesimal isometries of a hyperbolic metric with conical singularity of order $n-1$ at the origin. For $n=1$ this is the standard infinitesimal Möbius action, and for $n>1$ these vector fields have singularities at the origin and are incomplete, preventing integration to a global Lie group action. However, they naturally define an action Lie algebroid $\\mathcal{A}_n=\\mathfrak{sl}_2(\\mathbb{R})\\ltimes \\Dbarstar $ over the punctured disk. We construct an explicit Lie groupoid $\\mathcal{G}_n$ integrating $\\mathcal{A}_n$ and compare it to the \\v Severa--Weinstein groupoid. Although $\\mathcal{G}_n$ is not an action groupoid, its restriction to the boundary recovers an $n$-fold Möbius action on the boundary circle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30077",
  "author_total": 1,
  "authors": [
    {
      "name": "Rea Dalipi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6tygjer41m7sy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30077",
  "fulltext_url": null,
  "id": "6tygjer41m7sy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30077",
  "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.DG"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7204862818",
  "pdf_url": "https://arxiv.org/pdf/2608.30077",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6tygjer41m7sy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30077",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30077"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Lie groupoid integration of singular isometries of the Poincaré disk",
  "title_text": "Lie groupoid integration of singular isometries of the Poincaré disk",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/6tygjer41m7sy",
  "venue": "arXiv",
  "versions": []
}