{
  "id": "2205.15352",
  "version": "v1",
  "published": "2022-05-30T18:01:31.000Z",
  "updated": "2022-05-30T18:01:31.000Z",
  "title": "Canonical representations of surface groups",
  "authors": [
    "Aaron Landesman",
    "Daniel Litt"
  ],
  "comment": "56 pages, 3 figures. Comments welcome!",
  "categories": [
    "math.GT",
    "math.AG",
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $\\Sigma_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group of $\\Sigma_{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if $$\\rho: \\pi_1(\\Sigma_{g,n})\\to GL_r(\\mathbb{C})$$ is a representation whose conjugacy class has finite orbit under the mapping class group, and $r<\\sqrt{g+1}$, then $\\rho$ has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz. The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson's integrality conjecture for cohomologically rigid local systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-30T18:01:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34M56",
      "57K20",
      "14C30",
      "14H10",
      "11G99"
    ],
    "keywords": [
      "surface groups",
      "canonical representations",
      "mapping class group",
      "junho peter whang",
      "non-abelian hodge theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}