{
  "id": "1901.04463",
  "version": "v1",
  "published": "2019-01-14T18:55:47.000Z",
  "updated": "2019-01-14T18:55:47.000Z",
  "title": "Realizable ranks of joins and intersections of subgroups in free groups",
  "authors": [
    "Ignat Soroko"
  ],
  "comment": "25 pages, 20 figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "The famous Hanna Neumann Theorem gives an upper bound for the ranks of the intersection of arbitrary subgroups $H$ and $K$ of a non-abelian free group. It is an interesting question to \"quantify\" this bound with respect to the rank of $H\\vee K$, the subgroup generated by $H$ and $K$. We describe a set of realizable values $(rk(H\\vee K),rk(H\\cap K))$ for arbitrary $H$, $K$, and conjecture that this locus is complete. We study the combinatorial structure of Kent's topological pushout of the core graphs for $H$ and $K$, with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. Using it, we show that the conditions $rk(H\\vee K)=rk(H)+rk(K)-i$, $rk(H\\cap K)=\\frac{i(i-1)}2+1$, for $i\\ge 3$ are not realizable, thus resolving the remaining open case $m=4$ of Guzman's \"Group-Theoretic Conjecture\" in the affirmative, which in turn implies the validity of the corresponding \"Geometric Conjecture\" on hyperbolic $3$-manifolds with a $6$-free fundamental group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-14T18:55:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E05",
      "20E07",
      "20F65",
      "57M07"
    ],
    "keywords": [
      "realizable ranks",
      "intersection",
      "conjecture",
      "famous hanna neumann theorem",
      "free fundamental group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}