{
  "id": "2506.19195",
  "version": "v1",
  "published": "2025-06-23T23:35:21.000Z",
  "updated": "2025-06-23T23:35:21.000Z",
  "title": "Stallings' Group is Simply Connected at Infinity",
  "authors": [
    "Michael Mihalik"
  ],
  "comment": "18 pages 10 figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $F_2$ be the free group on two generators and let $B_n$ ($n \\geq 2$) denote the kernel of the homomorphism $$F_2 \\times \\cdots (n) \\cdots \\times F_2 \\rightarrow {\\mathbb Z}$$ sending all generators to the generator $1$ of $\\mathbb Z$. The groups $B_k$ are called the {\\it Bieri-Stallings} groups and $B_k$ is type $\\mathcal F_{k-1}$ but not $\\mathcal F_k$. For $n\\geq 3$ there are short exact sequences of the form $$1 \\rightarrow B_{n-1} \\rightarrow B_n \\rightarrow F_2 \\rightarrow 1.$$ This exact sequence can be used to show that $B_n$ is $(n-3)$-connected at infinity for $n\\geq 3$. Stallings' proved that $B_2$ is finitely generated but not finitely presented. We conjecture that for $n\\geq 2$, $B_n$ is $(n-2)$-connected at infinity. For $n=2$, this means that $B_2$ is 1-ended and for $n=3$ that $B_3$ (typically called Stallings' group) is simply connected at infinity. We verify the conjecture for $n=2$ and $n=3$. Our main result is the case $n=3$: Stalling's group is simply connected at $\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-23T23:35:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "short exact sequences",
      "stallings group",
      "main result",
      "free group",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}