{
  "id": "1608.00617",
  "version": "v1",
  "published": "2016-08-01T21:02:05.000Z",
  "updated": "2016-08-01T21:02:05.000Z",
  "title": "On rank of the join of two subgroups in a free group",
  "authors": [
    "Sergei V. Ivanov"
  ],
  "comment": "14 pages, 1 figure",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "Let $H, K$ be two finitely generated subgroups of a free group, let $\\langle H, K \\rangle$ denote the subgroup generated by $H, K$, called the join of $H, K$, and let neither of $H$, $K$ have finite index in $\\langle H, K \\rangle$. We prove the existence of an epimorphism $\\zeta : \\langle H, K \\rangle \\to F_2$, where $F_2$ is a free group of rank 2, such that the restriction of $\\zeta$ on both $H$ and $K$ is injective and the restriction $\\zeta_0 : H \\cap K \\to \\zeta (H) \\cap \\zeta (K) $ of $\\zeta$ on $H \\cap K $ to $\\zeta (H) \\cap \\zeta (K)$ is surjective. This is obtained as a corollary of an analogous result on rank of the generalized join of two finitely generated subgroups in a free group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-01T21:02:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E05",
      "20E07",
      "20F65",
      "57M07"
    ],
    "keywords": [
      "free group",
      "finitely generated subgroups",
      "finite index",
      "restriction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}