{
  "id": "1908.07014",
  "version": "v1",
  "published": "2019-08-19T18:30:58.000Z",
  "updated": "2019-08-19T18:30:58.000Z",
  "title": "Towards super-approximation in positive characteristic",
  "authors": [
    "Brian Longo",
    "Alireza Salehi Golsefidy"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\\rm GL}_{n_0}(\\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is a family of expanders under certain algebraic conditions. Here is a more precise formulation of our main result. For a positive integer $c_0$, we say a square-free polynomial is $c_0$-admissible if degree of irreducible factors of $f$ are distinct integers with prime factors at least $c_0$. Suppose $\\Omega$ is a finite symmetric subset of ${\\rm GL}_{n_0}(\\mathbb{F}_p(t))$, where $p$ is a prime more than $5$. Let $\\Gamma$ be the group generated by $\\Omega$. Suppose the Zariski-closure of $\\Gamma$ is connected, simply-connected, and absolutely almost simple; further assume that the field generated by the traces of ${\\rm Ad}(\\Gamma)$ is $\\mathbb{F}_p(t)$. Then for some positive integer $c_0$ the family of Cayley graphs ${\\rm Cay}(\\pi_{f(x)}(\\Gamma),\\pi_{f(x)}(\\Omega))$ as $f$ ranges in the set of $c_0$-admissible polynomials is a family of expanders, where $\\pi_{f(t)}$ is the quotient map for the congruence modulo $f(t)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-19T18:30:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E40",
      "20G30",
      "05C81"
    ],
    "keywords": [
      "positive characteristic",
      "super-approximation",
      "cayley graphs",
      "finite symmetric subset",
      "positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}