{
  "abstract": "We study representations of fundamental groups of closed orientable surfaces into \\(\\mathrm{SL}(2,\\mathbb{H})\\), whose projectivization \\(\\mathrm{PSL}(2,\\mathbb{H})\\) is the group of orientation-preserving isometries of real hyperbolic \\(5\\)-space, with the aim of developing a quaternionic analogue of Fenchel--Nielsen theory. We give a normal form for pairs of regular loxodromic elements with disjoint fixed point sets and show that, on an open dense generic locus, the corresponding moduli space is \\(15\\)-dimensional. We also associate fifteen word traces whose differentials are linearly independent on an open dense subset and hence give local real-analytic coordinates there. For a pair of pants with prescribed regular loxodromic boundary conjugacy classes, we show that the relative deformation space is locally \\(6\\)-dimensional. Combining this internal pants data with the three-dimensional boundary conjugacy data and the three-dimensional centralizer gluing freedom gives a local parameter decomposition for closed surface group representations. For a closed surface of genus \\(g\\), this yields \\(30g-30\\) real parameters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11438",
  "author_total": 3,
  "authors": [
    {
      "name": "Krishnendu Gongopadhyay"
    },
    {
      "name": "Sagar B. Kalane"
    },
    {
      "name": "Abhishek Mukherjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12geqp5h07aqb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11438",
  "fulltext_url": null,
  "id": "12geqp5h07aqb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11438",
  "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.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7212243244",
  "pdf_url": "https://arxiv.org/pdf/2609.11438",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12geqp5h07aqb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11438",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11438"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Trace Coordinates and Local Fenchel--Nielsen Parameters in Real Hyperbolic 5-Space",
  "title_text": "Trace Coordinates and Local Fenchel--Nielsen Parameters in Real Hyperbolic 5-Space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/12geqp5h07aqb",
  "venue": "arXiv",
  "versions": []
}