{
  "id": "0711.0091",
  "version": "v2",
  "published": "2007-11-01T10:58:44.000Z",
  "updated": "2009-06-18T04:41:30.000Z",
  "title": "Uniqueness of roots up to conjugacy for some affine and finite type Artin groups",
  "authors": [
    "Eon-Kyung Lee",
    "Sang-Jin Lee"
  ],
  "comment": "15 pages, 8 figures; version published by Math. Z",
  "journal": "Mathematische Zeitschrift, vol. 265, no. 3, pp. 571-587, 2010",
  "doi": "10.1007/s00209-009-0530-y",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "Let $G$ be one of the Artin groups of finite type ${\\mathbf B}_n={\\mathbf C}_n$, and affine type $\\tilde{\\mathbf A}_{n-1}$ and $\\tilde{\\mathbf C}_{n-1}$. In this paper, we show that if $\\alpha$ and $\\beta$ are elements of $G$ such that $\\alpha^k=\\beta^k$ for some nonzero integer $k$, then $\\alpha$ and $\\beta$ are conjugate in $G$. For the Artin group of type $\\mathbf A_n$, this was recently proved by J. Gonz\\'alez-Meneses. In fact, we prove a stronger theorem, from which the above result follows easily by using descriptions of those Artin groups as subgroups of the braid group on $n+1$ strands. Let $P$ be a subset of $\\{1,...,n\\}$. An $n$-braid is said to be \\emph{$P$-pure} if its induced permutation fixes each $i\\in P$, and \\emph{$P$-straight} if it is $P$-pure and it becomes trivial when we delete all the $i$-th strands for $i\\not\\in P$. Exploiting the Nielsen-Thurston classification of braids, we show that if $\\alpha$ and $\\beta$ are $P$-pure $n$-braids such that $\\alpha^k=\\beta^k$ for some nonzero integer $k$, then there exists a $P$-straight $n$-braid $\\gamma$ with $\\beta=\\gamma\\alpha\\gamma^{-1}$. Moreover, if $1\\in P$, the conjugating element $\\gamma$ can be chosen to have the first strand algebraically unlinked with the other strands. Especially in case of $P=\\{1,...,n\\}$, our result implies the uniqueness of root of pure braids, which was known by V. G. Bardakov and by D. Kim and D. Rolfsen.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-06-18T04:41:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20F10"
    ],
    "keywords": [
      "finite type artin groups",
      "uniqueness",
      "nonzero integer",
      "pure braids",
      "braid group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.0091L"
    }
  }
}