{
  "id": "2303.17429",
  "version": "v1",
  "published": "2023-03-30T14:51:32.000Z",
  "updated": "2023-03-30T14:51:32.000Z",
  "title": "Haagerup property and group-invariant percolation",
  "authors": [
    "Chiranjib Mukherjee",
    "Konstantin Recke"
  ],
  "categories": [
    "math.GR",
    "math.OA",
    "math.PR"
  ],
  "abstract": "Let $\\mathcal G$ be the Cayley graph of a finitely generated group $\\Gamma$. We show that $\\Gamma$ has the Haagerup property if and only if for every $\\alpha<1$, there is a $\\Gamma$-invariant bond percolation $\\mathbb P$ on $\\mathcal G$ with $\\mathbb E[\\mathrm{deg}_{\\omega}(g)]>\\alpha\\mathrm{deg}_{\\mathcal G}(g)$ for every vertex $g$ and with the two-point function $\\tau(g,h)=\\mathbb P[g\\leftrightarrow h]$ vanishing as $d(g,h)\\to\\infty$. Our result is inspired by the characterization of amenability by Benjamini, Lyons, Peres and Schramm [7]. To derive our result, we use the characterization of the Haagerup property in terms of actions on spaces with measured walls in the sense of Cherix, Martin and Valette [11]. Our proof is based on a new construction using invariant point processes on such spaces with measured walls, which leads to quantitative bounds on the two-point functions. These bounds yield in particular exponential decay of the two-point function in several examples, including co-compact Fuchsian groups and lamplighters over free groups. Moreover, our method allows us to strengthen a consequence of Kazhdan's property (T), due to Lyons and Schramm [44], to an {\\em equivalence}. Namely, we show that $\\Gamma$ has property (T) if and only if there exists a threshold $\\alpha^*<1$ such that for every $\\Gamma$-invariant bond percolation $\\mathbb P$ on $\\mathcal G$, $\\mathbb E[\\mathrm{deg}_\\omega(o)]>\\alpha^*\\mathrm{deg}(o)$ implies that the two-point function is bounded away from zero. We extend this result to the setting of {\\em relative} property (T). We then use the corresponding threshold to give a new proof of the fact, already observed by Gaboriau and Tucker-Drob [21], that there is no unique infinite cluster at the uniqueness threshold for Bernoulli bond percolation on Cayley graphs of groups admitting an infinite normal subgroup with relative property (T).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-30T14:51:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "haagerup property",
      "two-point function",
      "group-invariant percolation",
      "invariant bond percolation",
      "cayley graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}