{
  "abstract": "The signatures of $p$-rough paths form a subgroup of sufficiently high-level truncated tensor algebras, whose inverse limit is a subgroup of the full tensor algebra. For $p \\geq 1$, we provide a top-down description of the signature group as the inverse limit of finite-dimensional Carnot--Carathéodory geometries in the $p$-variation setting. We show that every compatible choice of metrics induces a topological tree structure on the inverse-limit group, under which the signature group is not a topological group. This extends the results of Enrico Le Donne and Roland Züst from bounded variation to rough paths. We also characterise the dependence of the inverse-limit groups and their metric completions on the choice of metric, identifying them with the tree-reduced path group of Horatio Boedihardjo, Xiang Geng, Terry Lyons, and Danyu Yang.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10875",
  "author_total": 3,
  "authors": [
    {
      "name": "Felix Medwed"
    },
    {
      "name": "Sylvie Paycha"
    },
    {
      "name": "Alexander Schmeding"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dk1jj741fnfpk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10875",
  "fulltext_url": null,
  "id": "dk1jj741fnfpk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10875",
  "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.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
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      "openalex",
      "2003|Stochastic processes and financial applications"
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  "openalex": "W7212299290",
  "pdf_url": "https://arxiv.org/pdf/2609.10875",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dk1jj741fnfpk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.10875",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10875"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Metric Geometry of the Signature Group for $p$-Variation Rough Paths",
  "title_text": "Metric Geometry of the Signature Group for p-Variation Rough Paths",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/dk1jj741fnfpk",
  "venue": "arXiv",
  "versions": []
}