{
  "id": "0804.0460",
  "version": "v1",
  "published": "2008-04-03T01:55:31.000Z",
  "updated": "2008-04-03T01:55:31.000Z",
  "title": "Algebro-Geometric Invariants of Finitely Generated Groups (The Profile of a Representation Variety)",
  "authors": [
    "S. Liriano S. Majewicz"
  ],
  "comment": "22 pages",
  "categories": [
    "math.GR",
    "math.AG"
  ],
  "abstract": "If G is a finitely generated group, and A an algebraic group, then Hom(G,A) is a possibly reducible algebraic variety denoted by R_A(G). Here we define the profile function, P_d(R_A(G)), of the representation variety of G over A to be P_d(R_A(G))=(N_d(R_A(G)),...,N_0(R_A(G))), where N_i(R_A(G)) stands for the number of irreducible components of R_A(G) of dimension i, where 0\\leq i\\leq d, and d=Dim(R_A(G)). We then use this invariant in the study of fg groups and prove various results. In particular, we show that if G an orientable surface group of genus g\\geq 1, then P_d(R_{SL(2,C)}(G))\\neq P_d(R_{PSL(2,C)}(G)). We also show that the same holds for G a torus knot group with presentation <x,y;x^p=y^t> where both p,t are greater than 2, and that the same also holds when G is a the fundamental group of a compact non-orientable surface of genus g\\geq 3. Further, we show that if a group G can be n+1 generated, and presented by <x_1,...,x_n,y ; W=y^p>, where W is a non-trivial word in F_n=<x_1,...,x_n>, and A=PSL(2, C), that then Dim(R_{A}(G)) is equal to Max{3n, Dim(R_{A}(G'))+2 \\} \\leq 3n+1, where G'=<x_1,...,x_n; W=1>. We also give a condition guaranteeing that the resulting algebraic variety is reducible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-04-03T01:55:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F29",
      "20F38"
    ],
    "keywords": [
      "finitely generated group",
      "representation variety",
      "algebro-geometric invariants",
      "reducible algebraic variety",
      "torus knot group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.0460M"
    }
  }
}