{
  "id": "2405.07688",
  "version": "v1",
  "published": "2024-05-13T12:23:02.000Z",
  "updated": "2024-05-13T12:23:02.000Z",
  "title": "Finitely generated groups and harmonic functions of slow growth",
  "authors": [
    "Mayukh Mukherjee",
    "Soumyadeb Samanta",
    "Soumyadip Thandar"
  ],
  "comment": "20 pages, comments most welcome! arXiv admin note: text overlap with arXiv:1505.01175 by other authors",
  "categories": [
    "math.GR",
    "math.MG",
    "math.PR"
  ],
  "abstract": "In this paper, we are mainly concerned with $(\\mathbb{G},\\mu)$-harmonic functions that grow at most polynomially, where $\\mathbb{G}$ is a finitely generated group with a probability measure $\\mu$. In the initial part of the paper, we focus on Lipschitz harmonic functions and how they descend onto finite index subgroups. We discuss the relations between Lipschitz harmonic functions and harmonic functions of linear growth and conclude that for groups of polynomial growth, they coincide. In the latter part of the paper, we specialise to positive harmonic functions and give a characterisation for strong Liouville property in terms of the Green's function. We show that the existence of a non-constant positive harmonic function of polynomial growth guarantees that the group cannot have polynomial growth.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-13T12:23:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finitely generated group",
      "slow growth",
      "lipschitz harmonic functions",
      "strong liouville property",
      "finite index subgroups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}