{
  "id": "math-ph/0503005",
  "version": "v3",
  "published": "2005-03-03T18:16:39.000Z",
  "updated": "2007-12-10T15:59:17.000Z",
  "title": "Existence of spectral gaps, covering manifolds and residually finite groups",
  "authors": [
    "Fernando Lledó",
    "Olaf Post"
  ],
  "comment": "final version (26 pages, 2 figures). to appear in Rev. Math. Phys",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the present paper we consider Riemannian coverings $(X,g) \\to (M,g)$ with residually finite covering group $\\Gamma$ and compact base space $(M,g)$. In particular, we give two general procedures resulting in a family of deformed coverings $(X,g_\\eps) \\to (M,g_\\eps)$ such that the spectrum of the Laplacian $\\Delta_{(X_\\eps,g_\\eps)}$ has at least a prescribed finite number of spectral gaps provided $\\eps$ is small enough. If $\\Gamma$ has a positive Kadison constant, then we can apply results by Br\\\"uning and Sunada to deduce that $\\spec \\Delta_{(X,g_\\eps)}$ has, in addition, band-structure and there is an asymptotic estimate for the number $N(\\lambda)$ of components of $\\spec {\\laplacian {(X,g_\\eps)}}$ that intersect the interval $[0,\\lambda]$. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-12-10T15:59:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J50",
      "35P15",
      "20E26",
      "57M10"
    ],
    "keywords": [
      "residually finite groups",
      "spectral gaps",
      "covering manifolds",
      "compact base space",
      "riemannian coverings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.ph...3005L"
    }
  }
}