{
  "id": "2303.03644",
  "version": "v2",
  "published": "2023-03-07T04:21:51.000Z",
  "updated": "2023-04-11T14:03:50.000Z",
  "title": "Quantifying separability in limit groups via representations",
  "authors": [
    "Keino Brown",
    "Olga Kharlampovich"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "We show that for any finitely generated subgroup $H$ of a limit group $L$ there exists a finite-index subgroup $K$ containing $H$, such that $K$ is a subgroup of a group obtained from $H$ by a series of extensions of centralizers and free products with $\\mathbb Z$. If $H$ is non-abelian, the $K$ is fully residually $H$. We also show that for any finitely generated subgroup of a limit group, there is a finite-dimensional representation of the limit group which separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a limit group. This generalizes the results of Louder, McReynolds and Patel. Another corollary is that a hyperbolic limit group satisfies the Geometric Hanna Neumann conjecture.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-04-11T14:03:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65"
    ],
    "keywords": [
      "quantifying separability",
      "finitely generated subgroup",
      "geometric hanna neumann conjecture",
      "hyperbolic limit group satisfies",
      "polynomial upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}