{
  "id": "1802.08996",
  "version": "v1",
  "published": "2018-02-25T12:56:45.000Z",
  "updated": "2018-02-25T12:56:45.000Z",
  "title": "Spectral gap property and strong ergodicity for groups of affine transformations of solenoids",
  "authors": [
    "Bachir Bekka",
    "Camille Francini"
  ],
  "comment": "17 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let X be a solenoid, that is, a compact finite dimensional connected abelian group with normalized Haar measure m, and let G be a countable discrete group acting on X by continuous affine transformations. We show that the probability measure preserving action of G on (X,m) does not have the spectral gap property if and only if there exists a p(G)-invariant proper subsolenoid Y of X such that the image of G in the affine group Aff(X/Y) of X/Y is a virtually solvable group, where p(G) is the automorphism part of G. When G is finitely generated or when X is a p-adic solenoid, the subsolenoid Y can be chosen so that the image of G in Aff(X/Y) is virtually abelian. In particular, an action of a group by affine transformations on a solenoid has the spectral gap property if and only if this action is strongly ergodic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-25T12:56:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22F10",
      "37A30"
    ],
    "keywords": [
      "spectral gap property",
      "affine transformations",
      "strong ergodicity",
      "finite dimensional connected abelian group",
      "compact finite dimensional connected abelian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}