{
  "id": "2101.04999",
  "version": "v1",
  "published": "2021-01-13T11:04:17.000Z",
  "updated": "2021-01-13T11:04:17.000Z",
  "title": "On arithmetic properties of solvable Baumslag-Solitar groups",
  "authors": [
    "Laurent Hayez",
    "Tom Kaiser",
    "Alain Valette"
  ],
  "comment": "16 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "For $0<\\alpha\\le 1$, we say that a sequence $(X_k)_{k>0}$ of $d$-regular graphs has property $D_\\alpha$ if there exists a constant $C>0$ such that $\\mathrm{diam}(X_k)\\ge C\\cdot|X_k|^\\alpha$. We investigate property $D_\\alpha$ for arithmetic box spaces of the solvable Baumslag-Solitar groups $BS(1,m)$ (with $m\\geq 2$): those are box spaces obtained by embedding $BS(1,m)$ into the upper triangular matrices in $GL_2(\\mathbb{Z}[1/m])$ and intersecting with a family $M_{N_k}$ of congruence subgroups of $GL_2(\\mathbb{Z}[1/m])$, where the levels $N_k$ are coprime with $m$ and $N_k|N_{k+1}$. We prove: - if an arithmetic box space has $D_\\alpha$, then $\\alpha\\le\\frac{1}{2}$~; - if the family $(N_k)_k$ of levels is supported on finitely many primes, the corresponding arithmetic box space has $D_{1/2}$~; - if the family $(N_k)_k$ of levels is supported on a family of primes with positive analytic primitive density, then the corresponding arithmetic box space does not have $D_\\alpha$, for every $\\alpha>0$. Moreover, we prove that if we embed $BS(1,m)$ in the group of invertible upper-triangular matrices $T_n(\\mathbb{Z}[1/m])$, then every finite index subgroup of the embedding contains a congruence subgroup. This is a version of the congruence subgroup property (CSP).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-13T11:04:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F16",
      "20F69"
    ],
    "keywords": [
      "solvable baumslag-solitar groups",
      "arithmetic properties",
      "corresponding arithmetic box space",
      "congruence subgroup property",
      "finite index subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}