{
  "abstract": "In any noncommutative Carnot group, for every $0\\le α<1$, we show the existence of a $C^{1,α}_\\mathrm H$ surface that is unrectifiable with respect to the Euclidean distance. Precisely, the surface has Euclidean Hausdorff dimension higher than its topological dimension. We show that in step-2 Carnot groups the construction gives hypersurfaces of maximal Euclidean Hausdorff dimension, equal to $q-\\frac{1+α}{2}$, where $q$ is the topological dimension of the group. Here the range of unrectifiability is sharp, since for $α=1$ the surface is rectifiable in the classical sense. In step-2 Carnot groups, it is also possible to prove that the $(q-\\frac12)$-dimensional Euclidean Hausdorff measure of every $C^1_\\mathrm H$ hypersurface is zero. These results extend the seminal work by Kirchheim and Serra Cassano in the first Heisenberg group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15851",
  "author_total": 2,
  "authors": [
    {
      "name": "Valentino Magnani"
    },
    {
      "name": "Jeremy Mirmina"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bjqezm935ee80.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15851",
  "fulltext_url": null,
  "id": "bjqezm935ee80",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15851",
  "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.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7221809386",
  "pdf_url": "https://arxiv.org/pdf/2610.15851",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bjqezm935ee80.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15851",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15851"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Unrectifiable surfaces with intrinsic regularity in Carnot groups",
  "title_text": "Unrectifiable surfaces with intrinsic regularity in Carnot groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/bjqezm935ee80",
  "venue": "arXiv",
  "versions": []
}