{
  "id": "1502.04339",
  "version": "v1",
  "published": "2015-02-15T17:58:11.000Z",
  "updated": "2015-02-15T17:58:11.000Z",
  "title": "Rigidity for group actions on homogeneous spaces by affine transformations",
  "authors": [
    "Mohamed Bouljihad"
  ],
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $\\Lambda$ a lattice in $G$, and $\\Gamma$ a subgroup of the affine group Aff$(G)$ stabilizing $\\Lambda$. Then the action of $\\Gamma$ on $G/\\Lambda$ has the rigidity property in the sense of S. Popa, if and only if the induced action of $\\Gamma$ on $\\mathbb{P}(\\frak{g})$ admits no $\\Gamma$-invariant probability measure, where $\\frak{g}$ is the Lie algebra of $G$. This generalizes results of M. Burger, and A. Ioana and Y. Shalom. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nilmanifolds associated to step 2 nilpotent Lie groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-15T17:58:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homogeneous spaces",
      "group actions",
      "affine transformations",
      "nilpotent lie groups",
      "real lie group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150204339B"
    }
  }
}