{
  "id": "1502.02413",
  "version": "v1",
  "published": "2015-02-09T10:07:53.000Z",
  "updated": "2015-02-09T10:07:53.000Z",
  "title": "Tilings of amenable groups",
  "authors": [
    "Tomasz Downarowicz",
    "Dawid Huczek",
    "Guohua Zhang"
  ],
  "comment": "23 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "We prove that for any infinite countable amenable group $G$, any $\\epsilon > 0$ and any finite subset $K\\subset G$, there exists a tiling (partition of $G$ into finite \"tiles\" using only finitely many \"shapes\"), where all the tiles are $(K; \\epsilon)$-invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of $G$ (in the sense that the mappings, associated to different from unity elements of $G$, have no fixpoints), on a zero-dimensional space, and which has topological entropy zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-09T10:07:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological entropy zero",
      "finite subset",
      "unity elements",
      "zero-dimensional space",
      "free action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}