{
  "id": "2303.10769",
  "version": "v1",
  "published": "2023-03-19T21:05:03.000Z",
  "updated": "2023-03-19T21:05:03.000Z",
  "title": "Ratio-limit boundaries of random walks on relatively hyperbolic groups",
  "authors": [
    "Adam Dor-On",
    "Matthieu Dussaule",
    "Ilya Gekhtman"
  ],
  "comment": "42 pages, 2 figures",
  "categories": [
    "math.GR",
    "math.DS",
    "math.GT",
    "math.OA",
    "math.PR"
  ],
  "abstract": "We study boundaries arising from limits of ratios of transition probabilities for random walks on relatively hyperbolic groups. We determine significant limitations of a strategy employed by Woess for computing the ratio-limit boundary in the hyperbolic case. On the one hand we employ results of the second and third authors to adapt this strategy to spectrally non-degenerate random walks, and show that the closure of minimal points in $R$-Martin boundary is the \\emph{unique} smallest invariant subspace in ratio-limit boundary. On the other hand we show that the general strategy can fail when the random walk is spectrally degenerate and adapted on a free product. Using our results, we are able to extend a theorem of the first author beyond the hyperbolic case and establish the existence of a co-universal quotient for Toeplitz C*-algebras arising from random walks which are spectrally non-degenerate on relatively hyperbolic groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-19T21:05:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J50",
      "20F67",
      "37A55",
      "37B05",
      "47L80"
    ],
    "keywords": [
      "relatively hyperbolic groups",
      "ratio-limit boundary",
      "hyperbolic case",
      "spectrally non-degenerate random walks",
      "determine significant limitations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}