{
  "id": "2407.07981",
  "version": "v1",
  "published": "2024-07-10T18:28:56.000Z",
  "updated": "2024-07-10T18:28:56.000Z",
  "title": "On the degree-two part of the associated graded of the lower central series of the Torelli group",
  "authors": [
    "Quentin Faes",
    "Gwenael Massuyeau",
    "Masatoshi Sato"
  ],
  "comment": "28 pages",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "We consider the associated graded $\\bigoplus_{k\\geq 1} \\Gamma_k \\mathcal{I} / \\Gamma_{k+1} \\mathcal{I} $ of the lower central series $\\mathcal{I} = \\Gamma_1 \\mathcal{I} \\supset \\Gamma_2 \\mathcal{I} \\supset \\Gamma_3 \\mathcal{I} \\supset \\cdots$ of the Torelli group $\\mathcal{I}$ of a compact oriented surface. Its degree-one part is well-understood by D. Johnson's seminal works on the abelianization of the Torelli group. The knowledge of the degree-two part $(\\Gamma_2 \\mathcal{I} / \\Gamma_3 \\mathcal{I})\\otimes \\mathbb{Q}$ with rational coefficients arises from works of S. Morita on the Casson invariant and R. Hain on the Malcev completion of $\\mathcal{I}$. Here, we prove that the abelian group $\\Gamma_2 \\mathcal{I} / \\Gamma_3 \\mathcal{I}$ is torsion-free, and we describe it as a lattice in a rational vector space. As an application, the group $\\mathcal{I}/\\Gamma_3 \\mathcal{I}$ is computed, and it is shown to embed in the group of homology cylinders modulo the surgery relation of $Y_3$-equivalence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-10T18:28:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower central series",
      "torelli group",
      "degree-two part",
      "rational coefficients arises",
      "johnsons seminal works"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}