{
  "abstract": "A finitely generated group acting locally elliptically on a complete \\CAT(0) space of finite geometric dimension fixes a point if distances to elliptic fixed sets are uniformly controlled by displacements. For arbitrary groups, the same hypothesis gives a fixed point in the space or its visual boundary. The proof combines a first-moment random-walk displacement estimate with Izeki's zero-drift theorem. For finite-rank $\\R$-buildings whose Euclidean metric factor has finite linear image, we obtain an all-type local-to-global theorem, including the nondiscrete $\\widetilde G_2$ case, with an explicit gap and without metric completeness; the linear-image restriction can be removed from the fixed-point conclusion when the Euclidean factor has dimension at most three. A tangent-cone criterion characterizes the gap and yields a cubical bound with optimal asymptotic growth $d/π$ in dimension $d$. For Hadamard manifolds, the gap is equivalent to uniformly bounded finite stabilizers in the isometry image, with stabilizer bounds of optimal polynomial growth in the gap constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41468",
  "author_total": 1,
  "authors": [
    {
      "name": "Jeroen Schillewaert"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3abzysrss73dp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41468",
  "fulltext_url": null,
  "id": "3abzysrss73dp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41468",
  "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.GR"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "arxiv",
      "math.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41468",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3abzysrss73dp.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41468",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41468"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A local-to-global fixed point theorem for $\\CAT(0)$ spaces with a uniform elliptic gap",
  "title_text": "A local-to-global fixed point theorem for CAT(0) spaces with a uniform elliptic gap",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/3abzysrss73dp",
  "venue": "arXiv",
  "versions": []
}