{
  "id": "2502.09513",
  "version": "v1",
  "published": "2025-02-13T17:29:30.000Z",
  "updated": "2025-02-13T17:29:30.000Z",
  "title": "On dynamics of the Mapping class group action on relative $\\text{PSL}(2,\\mathbb{R})$-Character Varieties",
  "authors": [
    "Ajay Kumar Nair"
  ],
  "comment": "27 pages, 10 figures",
  "categories": [
    "math.GT",
    "math.MG"
  ],
  "abstract": "In this paper, we study the mapping class group action on the relative $\\text{PSL}(2,\\mathbb{R})$-character varieties of punctured surfaces. It is well known that Minsky's primitive-stable representations form a domain of discontinuity for the $\\text{Out}(F_n)$-action on the $\\text{PSL}(2,\\mathbb{C})$-character variety. We define simple-stability of representations of fundamental group of a surface into $\\text{PSL}(2,\\mathbb{R})$ which is an analogue of the definition of primitive stability and prove that these representations form a domain of discontinuity for the $\\text{MCG}$-action. Our first main result shows that holonomies of hyperbolic cone surfaces are simple-stable. We also prove that holonomies of hyperbolic cone surfaces with exactly one cone-point of cone-angle less than $\\pi$ are primitive-stable, thus giving examples of an infinite family of indiscrete primitive-stable representations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-13T17:29:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mapping class group action",
      "character variety",
      "hyperbolic cone surfaces",
      "minskys primitive-stable representations form",
      "first main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}