{
  "id": "2206.13275",
  "version": "v1",
  "published": "2022-06-27T13:11:03.000Z",
  "updated": "2022-06-27T13:11:03.000Z",
  "title": "Cuts, flows and gradient conditions on harmonic functions",
  "authors": [
    "Antoine Gournay"
  ],
  "categories": [
    "math.GR",
    "math.RT"
  ],
  "abstract": "Reduced cohomology motivates to look at harmonic functions which satisfy certain gradient conditions. If $G$ is a direct product of two infinite groups or a (FC-central)-by-cyclic group, then there are no harmonic functions with gradient in $c_0$ on its Cayley graphs. From this, it follows that a metabelian group $G$ has no harmonic functions with gradient in $\\ell^p$. Furthermore, under a radial isoperimetric condition, groups whose isoperimetric profile is bounded by power of logarithms also have no harmonic functions with gradient in $\\ell^p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-27T13:11:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31C05",
      "22D10",
      "20J06",
      "43A15",
      "05C81",
      "20F69"
    ],
    "keywords": [
      "harmonic functions",
      "gradient conditions",
      "radial isoperimetric condition",
      "infinite groups",
      "reduced cohomology motivates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}