{
  "id": "2306.12533",
  "version": "v1",
  "published": "2023-06-21T19:38:03.000Z",
  "updated": "2023-06-21T19:38:03.000Z",
  "title": "On the intersection of fixed subgroups of $F_n\\times F_m$",
  "authors": [
    "André Carvalho"
  ],
  "comment": "11 pages, comments are welcome",
  "categories": [
    "math.GR"
  ],
  "abstract": "We prove that, although it is undecidable if a subgroup fixed by an automorphism intersects nontrivially an arbitrary subgroup of $F_n\\times F_m$, there is an algorithm that, taking as input a monomorphism and an endomorphism of $F_n\\times F_m$, decides whether their fixed subgroups intersect nontrivially. The general case of this problem, where two arbitrary endomorphisms are given as input remains unknown. We show that, when two endomorphisms of a certain type are given as input, this problem is equivalent to the Post Correspondence Problem for free groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-21T19:38:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "intersection",
      "input remains unknown",
      "post correspondence problem",
      "arbitrary subgroup",
      "free groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}