{
  "id": "1612.05152",
  "version": "v1",
  "published": "2016-12-15T17:32:04.000Z",
  "updated": "2016-12-15T17:32:04.000Z",
  "title": "Properness of nilprogressions and the persistence of polynomial growth of given degree",
  "authors": [
    "Romain Tessera",
    "Matthew Tointon"
  ],
  "comment": "34 pages",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "We show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of Bilu's result that a generalised arithmetic progression can be efficiently contained in a proper generalised arithmetic progression, and indeed an important ingredient in the proof is a version of Bilu's geometry-of-numbers argument carried out in a nilpotent Lie algebra. We also present some applications. We verify a conjecture of Benjamini that if $S$ is a symmetric generating set for a group such that $1\\in S$ and $|S^n|\\le Mn^D$ at some sufficiently large scale $n$ then $S$ exhibits polynomial growth of the same degree $D$ at all subsequent scales, in the sense that $|S^r|\\ll_{M,D}r^D$ for every $r\\ge n$. Our methods also provide an important ingredient in a forthcoming companion paper in which we show that if $(\\Gamma_n,S_n)$ is a sequence of Cayley graphs satisfying $|S_n^n|\\ll n^D$ as $n\\to\\infty$, and if $m_n\\gg n$ as $n\\to\\infty$, then every Gromov-Hausdorff limit of the sequence $(\\Gamma_{n},\\frac{d_{S_{n}}}{m_n})$ has homogeneous dimension bounded by $D$. We also note that our arguments imply that every approximate group has a large subset with a large quotient that is Freiman isomorphic to a subset of a torsion-free nilpotent group of bounded rank and step.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-15T17:32:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial growth",
      "important ingredient",
      "persistence",
      "properness",
      "proper generalised arithmetic progression"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}