{
  "id": "1311.2246",
  "version": "v3",
  "published": "2013-11-10T07:09:44.000Z",
  "updated": "2015-12-27T21:50:51.000Z",
  "title": "$Φ$-Harmonic Functions on Discrete Groups and First $\\ell^Φ$-Cohomology",
  "authors": [
    "Yaroslav Kopylov",
    "Roman Panenko"
  ],
  "comment": "13 pages",
  "journal": "Sib. Mat. J. 55, No. 5, 904--914 (2014)",
  "categories": [
    "math.GR"
  ],
  "abstract": "We study the first cohomology groups of a countable discrete group $G$ with coefficients in a $G$-module $\\ell^\\Phi(G)$, where $\\Phi$ is an $N$-function of class $\\Delta_2(0)\\cap \\nabla_2(0)$. In development of ideas of Puls and Martin--Valette, for a finitely generated group $G$, we introduce the discrete $\\Phi$-Laplacian and prove a theorem on the decomposition of the space of $\\Phi$-Dirichlet finite functions into the direct sum of the spaces of $\\Phi$-harmonic functions and $\\ell^\\Phi(G)$ (with an appropriate factorization). We also prove that if a finitely generated group $G$ has a finitely generated infinite amenable subgroup with infinite centralizer then $\\overline{H}^{1}(G,\\ell^{\\Phi}(G)) = 0$. In conclusion, we show the triviality of the first cohomology group for a wreath product of two groups one of which is nonamenable.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-15T09:42:47.000Z",
      "abstract": "We study the first cohomology of a (finitely generated) discrete group $G$ with coefficients in the left regular representation of $G$ in $\\ell^\\Phi(G)$ for an $N$-function $\\Phi$ belonging to the class $\\Delta_2(0)\\cap \\nabla_2(0)$. By analogy with Puls and Martin--Valette, we introduce the discrete $\\Phi$-Laplacian and prove a decomposition theorem for $\\Phi$-Dirichlet-finite functions. We also show that if $G$ contains an infinite normal amenable subgroup with infinite centralizer then the cohomology space $\\overline{H}^{1}(G,\\ell^{\\Phi}(G))=0$. Finally, we prove a theorem about the triviality of the first cohomology space for a wreath product of two groups the first of which is nonamenable.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-12-27T21:50:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20J06",
      "43A07",
      "43A15"
    ],
    "keywords": [
      "discrete group",
      "harmonic functions",
      "first cohomology group",
      "finitely generated group",
      "dirichlet finite functions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.2246K"
    }
  }
}