{
  "id": "2006.15705",
  "version": "v1",
  "published": "2020-06-28T20:43:53.000Z",
  "updated": "2020-06-28T20:43:53.000Z",
  "title": "Random walks on dense subgroups of locally compact groups",
  "authors": [
    "Michael Björklund",
    "Yair Hartman",
    "Hanna Oppelmayer"
  ],
  "comment": "40 pages, 0 figures. Comments are welcome!",
  "categories": [
    "math.DS",
    "math.GR",
    "math.PR"
  ],
  "abstract": "Let $\\Gamma$ be a countable discrete group, $H$ a lcsc totally disconnected group and $\\rho : \\Gamma \\rightarrow H$ a homomorphism with dense image. We develop a general and explicit technique which provides, for every compact open subgroup $L < H$ and bi-$L$-invariant probability measure $\\theta$ on $H$, a Furstenberg discretization $\\tau$ of $\\theta$ such that the Poisson boundary of $(H,\\theta)$ is a $\\tau$-boundary. Among other things, this technique allows us to construct examples of finitely supported random walks on certain lamplighter groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are prime, but not $L^p$-irreducible for any $p \\geq 1$, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we give an example of a countable discrete group $\\Gamma$ and two spread-out probability measures $\\tau_1$ and $\\tau_2$ on $\\Gamma$ such that the boundary entropy spectrum of $(\\Gamma,\\tau_1)$ is an interval, while the boundary entropy spectrum of $(\\Gamma,\\tau_2)$ is a Cantor set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-28T20:43:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "locally compact groups",
      "random walks",
      "dense subgroups",
      "boundary entropy spectrum",
      "countable discrete group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}