{
  "id": "2202.03877",
  "version": "v1",
  "published": "2022-02-08T14:03:53.000Z",
  "updated": "2022-02-08T14:03:53.000Z",
  "title": "Fuglede-Kadison determinants over free groups and Lehmer's constants",
  "authors": [
    "Fathi Ben Aribi"
  ],
  "comment": "18 pages, 2 figures, comments welcome. This paper is an expansion of the second half of the v3 of arXiv:2101.01678 [math.GT]",
  "categories": [
    "math.GR",
    "math.GT",
    "math.NT",
    "math.OA"
  ],
  "abstract": "Lehmer's famous problem asks whether the set of Mahler measures of polynomials with integer coefficients admits a gap at 1. In 2019, L\\\"uck extended this question to Fuglede-Kadison determinants of a general group, and he defined the Lehmer's constants of the group to measure such a gap. In this paper, we compute new values for Fuglede-Kadison determinants over non-cyclic free groups. As a consequence, we partially answer L\\\"uck's question, as we provide the new upper bound $\\frac{2}{\\sqrt{3}}$ for Lehmer's constants of all torsionfree groups which have non-cyclic free subgroups. Our proofs use relations between Fuglede-Kadison determinants and random walks on Cayley graphs, as well as works of Bartholdi and Dasbach-Lalin. Furthermore, we study several fundamental groups of small hyperbolic 3-manifolds, and we show that all of their Lehmer's constants are bounded by even smaller values than $\\frac{2}{\\sqrt{3}}$. For this, we use relations between Fuglede-Kadison determinants, $L^2$-torsions and hyperbolic volumes. Finally, we describe a method for computing upper approximations of Fuglede-Kadison determinants and we apply it for the group of the figure-eight knot complement.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-08T14:03:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R06",
      "47C15",
      "57K10",
      "57M05"
    ],
    "keywords": [
      "fuglede-kadison determinants",
      "lehmers constants",
      "non-cyclic free subgroups",
      "figure-eight knot complement",
      "non-cyclic free groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}