{
  "id": "2207.04294",
  "version": "v1",
  "published": "2022-07-09T16:17:10.000Z",
  "updated": "2022-07-09T16:17:10.000Z",
  "title": "Reidemeister classes in some wreath products by $\\mathbb Z^k$",
  "authors": [
    "Mikhail I. Fraiman",
    "Evgenij V. Troitsky"
  ],
  "categories": [
    "math.GR",
    "math.DS",
    "math.RT"
  ],
  "abstract": "Among restricted wreath products $G\\wr \\mathbb Z^k $, where $G$ is a finite Abelian group, we find three large classes of groups admitting an automorphism $\\varphi$ with finite Reidemeister number $R(\\varphi)$ (number of $\\varphi$-twisted conjugacy classes). In other words, groups from these classes do not have the $R_\\infty$ property. If a general automorphism $\\varphi$ of $G\\wr \\mathbb Z^k$ has a finite order (this is the case for $\\varphi$ detected in the first part of the paper) and $R(\\varphi)<\\infty$, we prove that $R(\\varphi)$ coincides with the number of equivalence classes of finite-dimensional irreducible unitary representations of $G\\wr \\mathbb Z^k$, which are fixed by the dual map $[\\rho]\\mapsto [\\rho\\circ \\varphi]$ (i.e. we prove the conjecture about finite twisted Burnside-Frobenius theorem, TBFT$_f$, for these $\\varphi$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-09T16:17:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E45",
      "22D10",
      "37C25"
    ],
    "keywords": [
      "reidemeister classes",
      "finite twisted burnside-frobenius theorem",
      "finite reidemeister number",
      "finite-dimensional irreducible unitary representations",
      "finite abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}