{
  "id": "1402.1481",
  "version": "v3",
  "published": "2014-02-06T20:46:54.000Z",
  "updated": "2016-05-03T11:51:40.000Z",
  "title": "Relative expanders",
  "authors": [
    "Goulnara Arzhantseva",
    "Romain Tessera"
  ],
  "comment": "24 pages, new title, Theorem 1.3 is new, more details in proofs of Lemma 2.5 and Theorem 7.3, final revised version",
  "journal": "Geometric and Functional Analysis (GAFA), 25 (2015), no. 2, 317-341",
  "doi": "10.1007/s00039-015-0316-9",
  "categories": [
    "math.GR",
    "math.FA",
    "math.MG"
  ],
  "abstract": "We exhibit a finitely generated group $G$ and a sequence of finite index normal subgroups $N_n\\trianglelefteq G$ such that for every finite generating subset $S\\subseteq G$, the sequence of finite Cayley graphs $(G/N_n, S)$ does not coarsely embed into any $L^p$-space for $1\\leqslant p<\\infty$ (moreover, into any uniformly curved Banach space), and yet admits no weakly embedded expander. The reason why our examples do not coarsely embed is a new phenomenon called relative expansion, which we define in terms of Poincar\\'e inequalities.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-19T16:55:09.000Z",
      "title": "Relatively expanding box spaces with no expansion",
      "abstract": "We exhibit a finitely generated group $G$ and a sequence of finite index normal subgroups $N_n\\trianglelefteq G$ such that for every finite generating subset $S\\subseteq G$, the sequence of finite Cayley graphs $(G/N_n, S)$ does not coarsely embed into any $L^p$-space for $1\\leqslant p<\\infty$ (moreover, into any uniformly curved Banach space), and yet admits no weakly embedded expander.",
      "comment": "22 pages, definition of weak embedding is clarified, appendix is added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2016-05-03T11:51:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B85",
      "20F69",
      "22D10",
      "20E22"
    ],
    "keywords": [
      "relatively expanding box spaces",
      "finite index normal subgroups",
      "finite cayley graphs",
      "finite generating subset",
      "uniformly curved banach space"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.1481A"
    }
  }
}