{
  "id": "1306.1542",
  "version": "v2",
  "published": "2013-06-06T20:19:10.000Z",
  "updated": "2015-02-13T14:03:35.000Z",
  "title": "Bounded cohomology with coefficients in uniformly convex Banach spaces",
  "authors": [
    "Mladen Bestvina",
    "Ken Bromberg",
    "Koji Fujiwara"
  ],
  "comment": "The title has been changed. The old title was \"Bounded cohomology via quasi-trees\". We prove a theorem for free groups (Theorem 1.1) using actions on trees, then deal with acylindrically hyperbolic groups using a work by Hull-Osin (Corollary 1.2). In the old version we had a direct proof using quasi-trees. We move the discussion on strongly contracting geodesics to a separate paper ([3])",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We show that for acylindrically hyperbolic groups $\\Gamma$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $\\rho$ of $\\Gamma$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(\\Gamma;\\rho)$ is infinite dimensional. The result was known for the regular representations on $\\ell^p(\\Gamma)$ with $1<p<\\infty$ by a different argument. But our result is new even for a non-abelian free group in this great generality for representations, and also the case for acylindrically hyperbolic groups follows as an application.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-06T20:19:10.000Z",
      "title": "Bounded cohomology via quasi-trees",
      "abstract": "We show that for the standard ``rank 1'' groups $\\Gamma$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $\\rho$ of $\\Gamma$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(\\Gamma;\\rho)$ is infinite dimensional. This is new even for representations $\\rho:G\\to O(n)$. The proof uses our earlier construction in the paper ``Constructing group actions on quasi-trees and applications to mapping class groups\" of group actions on quasi-trees.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-13T14:03:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bounded cohomology",
      "quasi-trees",
      "nontrivial finite normal subgroups",
      "arbitrary unitary representation",
      "uniformly convex banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.1542B"
    }
  }
}