{
  "id": "0909.1006",
  "version": "v2",
  "published": "2009-09-05T08:40:09.000Z",
  "updated": "2009-09-11T13:31:09.000Z",
  "title": "Lattices with and lattices without spectral gap",
  "authors": [
    "Bachir Bekka",
    "Alexander Lubotzky"
  ],
  "comment": "17 pages; corrected typos",
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "The following two results are shown. 1) Let $G$ be the $k$-rational points of a simple algebraic group over a local field $k$ and let $H$ be a lattice in $G.$ Then the regular representation of $G$ on $L^2(G/H)$ has a spectral gap (that is, there are no almost invariant unit vectors in the subspace of functions in $L^2(G/H)$ with zero mean). 2) There exist locally compact simple groups $G$ and lattices $H$ for which $L^2(G/H)$ has no spectral gap. This answers in the negative a question asked by Margulis. In fact, $G$ can be taken to be the group of orientation preserving automorphisms of a $k$-regular tree for $k>2.$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-09-11T13:31:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E40",
      "20E08"
    ],
    "keywords": [
      "spectral gap",
      "simple algebraic group",
      "locally compact simple groups",
      "invariant unit vectors",
      "regular tree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.1006B"
    }
  }
}