{
  "id": "1904.03907",
  "version": "v1",
  "published": "2019-04-08T09:31:20.000Z",
  "updated": "2019-04-08T09:31:20.000Z",
  "title": "Necessary conditions for tiling finitely generated amenable groups",
  "authors": [
    "Benjamin Hellouin de Menibus",
    "Hugo Maturana Cornejo"
  ],
  "comment": "12 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider a set of necessary conditions which are efficient heuristics for deciding when a set of Wang tiles cannot tile a group. Piantadosi gave a necessary and sufficient condition for the existence of a valid tiling of any free group. This condition is actually necessary for the existence of a valid tiling for an arbitrary finitely generated group. We then consider two other conditions: the first, also given by Piantadosi, is a necessary and sufficient condition to decide if a set of Wang tiles gives a strongly periodic tiling of the free group; the second, given by Chazottes et. al., is a necessary condition to decide if a set of Wang tiles gives a tiling of $\\mathbb Z^2$. We show that these last two conditions are equivalent. Joining and generalising approaches from both sides, we prove that they are necessary for having a valid tiling of any finitely generated amenable group, confirming a remark of Jeandel.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-08T09:31:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B50",
      "37B10",
      "05B45"
    ],
    "keywords": [
      "tiling finitely generated amenable groups",
      "necessary condition",
      "wang tiles",
      "valid tiling",
      "free group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}