{
  "id": "math/0701384",
  "version": "v1",
  "published": "2007-01-14T01:13:08.000Z",
  "updated": "2007-01-14T01:13:08.000Z",
  "title": "On character varieties, sets of discrete characters, and non-zero degree maps",
  "authors": [
    "Michel Boileau",
    "Steven Boyer"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "In this paper we use character variety methods to study homomorphisms between the fundamental groups of 3-manifolds, in particular those induced by non-zero degree maps. A {\\it knot manifold} is a compact, connected, irreducible, orientable 3-manifold whose boundary is an incompressible torus. A {\\it virtual epimorphism} is a homomorphism whose image is of finite index in its range. We show that the existence of such homomorphisms places constraints on the algebraic decomposition of a knot manifold's $PSL_2(\\mathbb C)$-character variety and consequently determine a priori bounds on the number of virtual epimorphisms between the fundamental groups of small knot manifolds with a fixed domain. In the second part of the paper we fix a small knot manifold $M$ and investigate various sets of characters of representations with discrete image in $PSL_2(\\mathbb C)$. The topology of these sets is intimately related to the algebraic structure of the $PSL_2(\\mathbb C)$-character variety of $M$ as well as dominations of manifolds by $M$ and its Dehn fillings. In particular, we apply our results to study families of non-zero degree maps $f_n: M(\\alpha_n) \\to V_n$ where $M(\\alpha_n)$ is the $\\alpha_n$-Dehn filling of $M$ and $V_n$ is either a hyperbolic manifold or $\\widetilde{SL_2}$ manifold. We show that quite often, up to taking a subsequence, there is a knot manifold $V$, slopes $\\beta_j$ on $\\partial V$ such that $V_j \\cong V(\\beta_j)$, and a non-zero degree map $M \\to V$ which induces $f_j$ up to homotopy. The work of the first part of the paper is then applied to construct infinite families of small, closed, connected, orientable 3-manifolds which do not admit non-zero degree maps, other than homeomorphisms, to any hyperbolic manifold, or even manifolds with infinite fundamental groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-01-14T01:13:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M05",
      "57M50",
      "57R65"
    ],
    "keywords": [
      "discrete characters",
      "small knot manifold",
      "fundamental groups",
      "hyperbolic manifold",
      "admit non-zero degree maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1384B"
    }
  }
}