{
  "id": "math/0410275",
  "version": "v2",
  "published": "2004-10-11T15:41:47.000Z",
  "updated": "2005-03-14T15:28:33.000Z",
  "title": "Palindromes and orderings in Artin groups",
  "authors": [
    "Florian Deloup"
  ],
  "comment": "16 pages, 4 figures. Main result extended to Artin groups. simplification of classification of $\\tau$-invariant palindromes in finite Artin groups. Added references",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "The braid group $B_{n}$, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism ${\\rm{rev}}: B_{n} \\to B_{n}$, $v \\mapsto \\bar{v}$, defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation $\\tau:x \\mapsto \\Delta^{-1}x \\Delta$ by the generalized half-twist (Garside element). More generally, the involution ${\\rm{rev}}$ is defined for all Artin groups (equipped with Artin's presentation) and the involution $\\tau$ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We classify palindromes and palindromes invariant under $\\tau$ in Artin groups of finite type. The tools are elementary rewriting and the construction of explicit left-orderings compatible with rev. Finally, we discuss generalizations to Artin groups of infinite type and Garside groups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-03-14T15:28:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36"
    ],
    "keywords": [
      "artin groups",
      "artins presentation",
      "involution",
      "explicit left-orderings",
      "palindromes invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10275D"
    }
  }
}