{
  "abstract": "We develop Picard Farey symbols for finite-index subgroups of \\(G=\\PSL_2(\\ZZ[i])\\) from the Gaussian Farey tessellation of \\(\\HH^3\\) by ideal octahedra. The ambient geometry naturally involves two different index-two extensions of \\(G\\): the orientation-preserving group \\(\\PGL_2(\\ZZ[i])\\) and a reflection extension. This distinction explains the two \\(G\\)-orbits of Farey faces, which merge into one \\(\\PGL_2\\)-orbit, and the classical extended modular tessellation appears as a totally geodesic slice. A Picard Farey symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and cooriented face-pairing transformations. We prove that such a symbol reconstructs the finite Picard-cell action and hence the subgroup; moreover, when subgroup membership is decidable, the symbol can be found by a terminating finite double-coset procedure. For torsion-free subgroups we obtain whole-octahedron fundamental domains. The marked symbol also supplies finite orbit relations for the established Gaussian Steinberg edge--face presentation, with its corresponding Bianchi modular-symbol relations. When the natural modular-to-Picard coset map is bijective, a side-connected Kulkarni special polygon lifts, using the same modular representatives, to a Picard fundamental polyhedron whose intersection with the invariant real plane is exactly that polygon; the level-$(2+i)$/level-$5$ family is worked out explicitly. Comparison examples show that octahedral boundary minimization, absolute geometric face minimization, and preservation of such a modular section are distinct problems. Canonical reduction and effective Hecke reduction remain open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36629",
  "author_total": 2,
  "authors": [
    {
      "name": "Devendra Tiwari"
    },
    {
      "name": "Helena Verrill"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bd3fzemx5awkn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36629",
  "fulltext_url": null,
  "id": "bd3fzemx5awkn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36629",
  "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.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214917937",
  "pdf_url": "https://arxiv.org/pdf/2609.36629",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bd3fzemx5awkn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2609.36629",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36629"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Farey Symbols for the Picard Group",
  "title_text": "Farey Symbols for the Picard Group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/bd3fzemx5awkn",
  "venue": "arXiv",
  "versions": []
}