{
  "id": "1503.06473",
  "version": "v1",
  "published": "2015-03-22T20:31:54.000Z",
  "updated": "2015-03-22T20:31:54.000Z",
  "title": "Local spectral gap in simple Lie groups and applications",
  "authors": [
    "Rémi Boutonnet",
    "Adrian Ioana",
    "Alireza Salehi Golsefidy"
  ],
  "categories": [
    "math.GR",
    "math.CO",
    "math.DS"
  ],
  "abstract": "We introduce a novel notion of {\\it local spectral gap} for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $\\Gamma\\curvearrowright G$, whenever $\\Gamma$ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group $G$. This extends to the non-compact setting recent works of Bourgain and Gamburd \\cite{BG06,BG10}, and Benoist and de Saxc\\'{e} \\cite{BdS14}. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on $G$. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique $\\Gamma$-invariant finitely additive measure defined on all bounded measurable subsets of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-22T20:31:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "applications",
      "arbitrary connected simple lie group",
      "finitely additive measure",
      "left translation action",
      "establish local spectral gap"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150306473B"
    }
  }
}