{
  "id": "1503.02050",
  "version": "v1",
  "published": "2015-03-06T20:05:42.000Z",
  "updated": "2015-03-06T20:05:42.000Z",
  "title": "Finite group extensions of shifts of finite type: K-theory, Parry and Livšic",
  "authors": [
    "Mike Boyle",
    "Scott Schmieding"
  ],
  "comment": "40 pages",
  "categories": [
    "math.DS",
    "math.KT"
  ],
  "abstract": "This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group G, Parry showed how to define a G-extension S_A from a square matrix A over Z_+G, and classified the extensions up to topological conjugacy by the strong shift equivalence class of A over Z_+G. Parry asked in this case if the det(I-tA) (which captures the \"periodic data\" of the extension) would classify up to finitely many topological conjugacy classes the extensions by G of a fixed mixing shift of finite type. When the algebraic K-theory group NK_1(ZG) is nontrivial (e.g., for G=Z/4), we show the dynamical zeta function for any such extension is consistent with infinitely many topological conjugacy classes. Independent of NK_1(ZG): for every nontrivial abelian G we show there exists a shift of finite type with an infinite family of mixing nonconjugate G extensions with the same dynamical zeta function. We define computable complete invariants for the periodic data of the extension for G not necessarily abelian, and extend all the above results to the nonabelian case. There is other work on basic invariants. The constructions require the \"positive K-theory\" setting for positive equivalence of matrices over ZG[t].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-06T20:05:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B10",
      "19M05"
    ],
    "keywords": [
      "finite type",
      "topological conjugacy classes",
      "dynamical zeta function",
      "strong shift equivalence class",
      "periodic data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150302050B"
    }
  }
}