{
  "aliases": null,
  "backlinks": [
    {
      "field": "specializes",
      "target": "6D1DV09D6P"
    },
    {
      "field": "specializes",
      "target": "BNR4H5GCTP"
    },
    {
      "field": "defines",
      "target": "8QE86W4GQ6"
    },
    {
      "field": "uses",
      "target": "SRS8WT4H8Y"
    }
  ],
  "body": "\n## Summary\n\nAn Artin group, or Artin–Tits group, is a [[group|G5S1M9]] defined by a Coxeter matrix.\nEach pair of generators satisfies a braid relation of a given length, or no relation.\nAdding the relations $s_i^2 = 1$ turns it into the corresponding [[Coxeter group|G6FND7X69J]].\nThe [[braid groups|BNR4H5GCTP]], the [[free groups|0KTW26C80J]], the free abelian groups and the [[right-angled Artin groups|6D1DV09D6P]] are Artin groups.\nMany basic questions about general Artin groups, such as whether they are torsion-free, are still open.\n\n## Explanations\n\n### L1 human\n\nA braid relation says that two generators can be pushed past each other in a fixed pattern.\nFor example, $aba = bab$ is the relation of two neighbouring crossings in a braid with three strands.\nAn Artin group chooses, for each pair of generators, how long the alternating words in such a relation are, or imposes no relation at all.\nIt is like a Coxeter group without the rule that each generator squares to the identity, so the generators have infinite order.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "20F36"
    ],
    "name": "Artin group",
    "same_as": {
      "wikidata": "Q3956993",
      "wikipedia": "Artin–Tits group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Artin%E2%80%93Tits_group"
      }
    ],
    "specializes": [
      "G5S1M9"
    ],
    "tag": "SAWBDBHE41"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/SAWBDBHE41/",
  "links": [
    {
      "field": "specializes",
      "target": "G5S1M9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/SAWBDBHE41.md",
  "msc": [
    "20F36"
  ],
  "name": "Artin group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SAW/SAWBDBHE41.md",
  "status": null,
  "tag": "SAWBDBHE41",
  "uri": "m2://SAWBDBHE41",
  "url": "https://math2.tamnd.com/wiki/artin-group/",
  "writing": {
    "tag": "SAWBDBHE41",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}