{
  "id": "0711.4238",
  "version": "v2",
  "published": "2007-11-27T12:27:39.000Z",
  "updated": "2009-04-20T10:25:31.000Z",
  "title": "Infinite groups with fixed point properties",
  "authors": [
    "G. Arzhantseva",
    "M. R. Bridson",
    "T. Januszkiewicz",
    "I. J. Leary",
    "A. Minasyan",
    "J. Swiatkowski"
  ],
  "comment": "Version 2: 29 pages. This is the final published version of the article",
  "journal": "Geometry & Topology 13 (2009), no. 3, 1229-1263",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We construct finitely generated groups with strong fixed point properties. Let $\\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any action of $Q$ on any $X\\in \\mathcal{X}_{ac}$, there is a global fixed point. Moreover, $Q$ may be chosen to be simple and to have Kazhdan's property (T). We construct a finitely presented infinite group $P$ that admits no non-trivial action by diffeomorphisms on any smooth manifold in $\\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\\geq 1$ and each prime $p$, we construct a non-elementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$-group.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-04-20T10:25:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67",
      "57Sxx"
    ],
    "keywords": [
      "infinite group",
      "non-elementary hyperbolic group",
      "strong fixed point properties",
      "hausdorff spaces",
      "smooth manifold"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.4238A"
    }
  }
}