{
  "id": "2410.16002",
  "version": "v1",
  "published": "2024-10-21T13:34:38.000Z",
  "updated": "2024-10-21T13:34:38.000Z",
  "title": "Profinite almost rigidity in 3-manifolds",
  "authors": [
    "Xiaoyu Xu"
  ],
  "comment": "50 pages, comments are welcome!",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "We prove that any compact, orientable 3-manifold with empty or incompressible toral boundary is profinitely almost rigid among all compact, orientable, boundary-incompressible 3-manifolds, i.e. the profinite completion of its fundamental group determines its homeomorphism type up to finitely many possibilities. Moreover, the profinite completion of the fundamental group of a mixed 3-manifold, together with the peripheral structure, uniquely determines the homeomorphism type of its Seifert part (i.e. the maximal graph manifold components in the JSJ-decomposition). On the other hand, without assigning the peripheral structure, the profinite completion of a mixed 3- manifold group may not even determine the fundamental group of its Seifert part. The proof is based on JSJ-decomposition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-21T13:34:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M05",
      "20E18"
    ],
    "keywords": [
      "profinite completion",
      "maximal graph manifold components",
      "seifert part",
      "peripheral structure",
      "homeomorphism type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}