{
  "id": "2012.12197",
  "version": "v1",
  "published": "2020-12-22T17:37:47.000Z",
  "updated": "2020-12-22T17:37:47.000Z",
  "title": "On the spread of infinite groups",
  "authors": [
    "Charles Garnet Cox"
  ],
  "comment": "13 pages. Comments welcome!",
  "categories": [
    "math.GR"
  ],
  "abstract": "A group is $\\frac32$-generated if every non-trivial element is part of a generating pair. In 2019 Donoven and Harper showed that many Thompson groups are $\\frac32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\\frac32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups of the Houghton group FSym$(\\mathbb{Z})\\rtimes\\mathbb{Z}$. We then show that the first two groups in our family are $\\frac32$-generated, and investigate the related notion of spread for these groups. We are able to show that they have finite spread which is greater than 2. Other than $\\mathbb{Z}$ and the Tarski monsters, neither of these properties have been proven for any infinite group before.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-22T17:37:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F05"
    ],
    "keywords": [
      "infinite group",
      "houghton group fsym",
      "finite index subgroups",
      "proper quotient cyclic",
      "tarski monsters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}