{
  "id": "2501.11363",
  "version": "v1",
  "published": "2025-01-20T09:50:03.000Z",
  "updated": "2025-01-20T09:50:03.000Z",
  "title": "Boundedness of diffeomorphism groups of manifold pairs -- Circle case --",
  "authors": [
    "Kazuhiko Fukui",
    "Tatsuhiko Yagasaki"
  ],
  "comment": "34 pages",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "In this paper we study boundedness of conjugation invariant norms on diffeomorphism groups of manifold pairs. For the diffeomorphism group ${\\mathcal D} \\equiv {\\rm Diff}(M,N)_0$ of a closed manifold pair $(M, N)$ with $\\dim N \\geq 1$, first we clarify the relation among the fragmentation norm, the conjugation generated norm, the commutator length $cl$ and the commutator length with support in balls $clb$ and show that ${\\mathcal D}$ is weakly simple relative to a union of some normal subgroups of ${\\mathcal D}$. For the boundedness of these norms, this paper focuses on the case where $N$ is a union of $m$ circles. In this case, the rotation angle on $N$ induces a quasimorphism $\\nu : {\\rm Isot}(M, N)_0 \\to {\\Bbb R}^m$, which determines a subgroup $A$ of ${\\Bbb Z}^m$ and a function $\\widehat{\\nu} : {\\mathcal D} \\to {\\Bbb R}^m/A$. If ${\\rm rank}\\,A = m$, these data leads to an upper bound of $clb$ on ${\\mathcal D}$ modulo the normal subgroup ${\\mathcal G} \\cong {\\rm Diff}_c(M - N)_0$. Then, some upper bounds of $cl$ and $clb$ on ${\\mathcal D}$ are obtained from those on ${\\mathcal G}$. As a consequence, the group ${\\mathcal D}$ is uniformly weakly simple and bounded when $\\dim M \\neq 2,4$. On the other hand, if ${\\rm rank}\\,A < m$, then the group ${\\mathcal D}$ admits a surjective quasimorphism, so it is unbounded and not uniformly perfect. We examine the group $A$ in some explicit examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-20T09:50:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R50",
      "57R52",
      "37C05"
    ],
    "keywords": [
      "diffeomorphism group",
      "circle case",
      "commutator length",
      "upper bound",
      "normal subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}