{
  "id": "1106.2623",
  "version": "v1",
  "published": "2011-06-14T07:03:11.000Z",
  "updated": "2011-06-14T07:03:11.000Z",
  "title": "On the spectral theory of groups of affine transformations of compact nilmanifolds",
  "authors": [
    "Bachir Bekka",
    "Yves Guivarc'h"
  ],
  "comment": "52 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $N$ be a connected and simply connected nilpotent Lie group, $\\Lambda$ a lattice in $N$, and $X=N/\\Lambda$ the corresponding nilmanifold. Let $Aff(X)$ be the group of affine transformations of $X$. We characterize the countable subgroups $H$ of $Aff(X)$ for which the action of $H$ on $X$ has a spectral gap, that is, such that the associated unitary representation $U$ of $H$ on the space of functions from $L^2(X)$ with zero mean does not weakly contain the trivial representation. Denote by $T$ the maximal torus factor associated to $X$. We show that the action of $H$ on $X$ has a spectral gap if and only if there exists no proper $H$-invariant subtorus $S$ of $T$ such that the projection of $H$ on $Aut (T/S)$ has an abelian subgroup of finite index. We first establish the result in the case where $X$ is a torus. In the case of a general nilmanifold, we study the asymptotic behaviour of matrix coefficients of $U$ using decay properties of metaplectic representations of symplectic groups. The result shows that the existence of a spectral gap for subgroups of $Aff(X)$ is equivalent to strong ergodicity in the sense of K.Schmidt. Moreover, we show that the action of $H$ on $X$ is ergodic (or strongly mixing) if and only if the corresponding action of $H$ on $T$ is ergodic (or strongly mixing).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-06-14T07:03:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22F30",
      "37A30"
    ],
    "keywords": [
      "affine transformations",
      "spectral theory",
      "compact nilmanifolds",
      "spectral gap",
      "simply connected nilpotent lie group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.2623B"
    }
  }
}