{
  "id": "2310.10401",
  "version": "v1",
  "published": "2023-10-16T13:42:51.000Z",
  "updated": "2023-10-16T13:42:51.000Z",
  "title": "Representations of braid groups via cyclic covers of the sphere: Zariski closure and arithmeticity",
  "authors": [
    "Gabrielle Menet",
    "Duc-Manh Nguyen"
  ],
  "comment": "48 pages, comments welcome",
  "categories": [
    "math.GT",
    "math.CV",
    "math.GR"
  ],
  "abstract": "Let $d \\geq 2$ and $n\\geq 3$ be two natural numbers. Given any sequence $\\kappa=(k_1,\\dots,k_n) \\in \\mathbb{Z}^n$ such that $\\gcd(k_1,\\dots,k_n,d)=1$, we consider the family of Riemann surfaces obtained from the plane curves defined by $y^d=\\prod_{i=1}^n(x-b_i)^{k_i}$, where $\\{b_1,\\dots,b_n\\}$ are $n$ distinct points in $\\mathbb{C}$. The monodromy of the fiber cohomology of this family provides us with a representation of the pure braid group $\\mathrm{PB}_n$ into some symplectic group. By restricting to a specific subspace in the cohomology of the fiber, we obtain a representation $\\rho_d$ of $\\mathrm{PB}_n$ into a linear algebraic group defined over $\\mathbb{Q}$. The first main result of this paper is a criterion for the Zariski closure of the image of $\\rho_d$ to be maximal, and the second main result is a criterion for the image to be an arithmetic lattice in the target group. The latter generalizes previous results of Venkataramana and gives an answer to a question by McMullen.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-16T13:42:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K20",
      "20C99"
    ],
    "keywords": [
      "zariski closure",
      "cyclic covers",
      "representation",
      "arithmeticity",
      "second main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}