{
  "id": "1004.5308",
  "version": "v2",
  "published": "2010-04-29T14:41:17.000Z",
  "updated": "2011-01-25T06:36:46.000Z",
  "title": "Periodic elements in Garside groups",
  "authors": [
    "Eon-Kyung Lee",
    "Sang-Jin Lee"
  ],
  "comment": "The contents of the 8-page paper \"Notes on periodic elements of Garside groups\" (arXiv:0808.0308) have been subsumed into this version. 27 pages",
  "journal": "Journal of Pure and Applied Algebra, vol. 215, no. 10, pp. 2295-2314, 2011",
  "doi": "10.1016/j.jpaa.2010.12.011",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "Let $G$ be a Garside group with Garside element $\\Delta$, and let $\\Delta^m$ be the minimal positive central power of $\\Delta$. An element $g\\in G$ is said to be 'periodic' if some power of it is a power of $\\Delta$. In this paper, we study periodic elements in Garside groups and their conjugacy classes. We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of $G$ is cyclic; if $g^k=\\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\\Delta^a$; every finite subgroup of the quotient group $G/<\\Delta^m>$ is cyclic. By a classical theorem of Brouwer, Ker\\'ekj\\'art\\'o and Eilenberg, an $n$-braid is periodic if and only if it is conjugate to a power of one of two specific roots of $\\Delta^2$. We generalize this to Garside groups by showing that every periodic element is conjugate to a power of a root of $\\Delta^m$. We introduce the notions of slimness and precentrality for periodic elements, and show that the super summit set of a slim, precentral periodic element is closed under any partial cycling. For the conjugacy problem, we may assume the slimness without loss of generality. For the Artin groups of type $A_n$, $B_n$, $D_n$, $I_2(e)$ and the braid group of the complex reflection group of type $(e,e,n)$, endowed with the dual Garside structure, we may further assume the precentrality.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-01-25T06:36:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20F10"
    ],
    "keywords": [
      "garside group",
      "minimal positive central power",
      "complex reflection group",
      "dual garside structure",
      "precentral periodic element"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.5308L"
    }
  }
}