{
  "id": "0712.3104",
  "version": "v1",
  "published": "2007-12-19T06:22:41.000Z",
  "updated": "2007-12-19T06:22:41.000Z",
  "title": "Orbit decidability and the conjugacy problem for some extensions of groups",
  "authors": [
    "O. Bogopolski",
    "A. Martino",
    "E. Ventura"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Given a short exact sequence of groups with certain conditions, $1\\to F\\to G\\to H\\to 1$, we prove that $G$ has solvable conjugacy problem if and only if the corresponding action subgroup $A\\leqslant Aut(F)$ is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form $\\mathbb{Z}^2\\rtimes F_m$, $F_2\\rtimes F_m$, $F_n \\rtimes \\mathbb{Z}$, and $\\mathbb{Z}^n \\rtimes_A F_m$ with virtually solvable action group $A\\leqslant GL_n(\\mathbb{Z})$. Also, we give an easy way of constructing groups of the form $\\mathbb{Z}^4\\rtimes F_n$ and $F_3\\rtimes F_n$ with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in $Aut(F_2)$ is given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-12-19T06:22:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F10",
      "20F28"
    ],
    "keywords": [
      "orbit decidability",
      "extensions",
      "short exact sequence",
      "virtually solvable action group",
      "unsolvable conjugacy problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.3104B"
    }
  }
}