{
  "id": "1911.01893",
  "version": "v1",
  "published": "2019-11-05T15:56:15.000Z",
  "updated": "2019-11-05T15:56:15.000Z",
  "title": "Classifying spaces for chains of families of subgroups",
  "authors": [
    "Víctor Moreno"
  ],
  "comment": "The author's PhD thesis, 106 pages",
  "categories": [
    "math.GR",
    "math.AT"
  ],
  "abstract": "This thesis concerns the study of the Bredon cohomological and geometric dimensions of a discrete group $G$ with respect to a family $\\mathfrak{F}$ of subgroups of $G$. With that purpose, we focus on building finite-dimensional models for $\\operatorname{E}_{\\mathfrak{F}} \\left( G \\right)$. The cases of the family $\\mathfrak{Fin}$ of finite subgroups of a group and the family $\\mathfrak{VC}$ of virtually cyclic subgroups of a group have been widely studied and many tools have been developed to relate the classifying spaces for $\\mathfrak{VC}$ with those for $\\mathfrak{Fin}$. Given a discrete group $G$ and an ascending chain $\\mathfrak{F}_0 \\subseteq \\mathfrak{F}_1 \\subseteq \\ldots \\subseteq \\mathfrak{F}_n \\subseteq \\ldots$ of families of subgroups of $G$, we provide a recursive methodology to build models for $\\operatorname{E}_{\\mathfrak{F}_r} \\left( G \\right)$ and give certain conditions under which the models obtained are finite-dimensional. We provide upper bounds for both the Bredon cohomological and geometric dimensions of $G$ with respect to the families $\\left(\\mathfrak{F}_r\\right)_{r\\in\\mathbb{N}}$ utilising the classifying spaces obtained. We consider then the families $\\mathfrak{H}_r$ of virtually polycyclic subgroups of Hirsch length less than or equal to $r$, for $r\\in\\mathbb{N}$. We apply the results obtained for chains of families of subgroups to the chain $\\mathfrak{H}_0 \\subseteq \\mathfrak{H}_1 \\subseteq \\ldots$ for an arbitrary virtually polycyclic group $G$, proving that the corresponding Bredon dimensions are both bounded above by $h(G) + r$, where $h(G)$ is the Hirsch length of $G$. Finally, we give similar results for the same chain of families of subgroups and an arbitrary locally virtually polycyclic group as the ambient group, obtaining in this case the upper bound $h(G) + r + 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-05T15:56:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55R35",
      "20J06",
      "18G99"
    ],
    "keywords": [
      "classifying spaces",
      "geometric dimensions",
      "upper bound",
      "discrete group",
      "hirsch length"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 106,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}