{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $A$ be a finite [[generating set|0RP7VWQ0YN]] of $G$ closed under inverses.\nFor a word $w$ over $A$, write $\\bar w$ for the element of $G$ it represents.\nLet $\\$$ be a new padding symbol.\nFor words $u, v$ over $A$, write $u$ and $v$ one above the other and fill the end of the shorter one with $\\$$.\nThe result is a word over $(A \\cup \\set{\\$})^2$, the padded pair $(u, v)^{\\$}$.\nAn *automatic structure* on $G$ with respect to $A$ consists of finite automata $W$ over $A$ and $M_x$ over $(A \\cup \\set{\\$})^2$ for $x \\in A \\cup \\set{e}$, such that:\n\n1. Every element of $G$ equals $\\bar w$ for at least one word $w$ accepted by $W$.\n2. For each $x \\in A \\cup \\set{e}$ and all words $u, v$ accepted by $W$, the automaton $M_x$ accepts $(u, v)^{\\$}$ if and only if $\\bar u \\, x = \\bar v$ in $G$.\n\nThe group $G$ is *automatic* if it has an automatic structure with respect to some finite generating set.\n\n## Remarks\n\nEquivalently, $G$ is automatic if there is a regular language $L$ over $A$ mapping onto $G$ with the fellow traveller property.\nThis means there is a constant $k$ such that whenever $u, v \\in L$ and $\\bar u^{-1} \\bar v \\in A \\cup \\set{e}$, the prefixes $u(t)$ and $v(t)$ of length $t$ satisfy $d_A(\\overline{u(t)}, \\overline{v(t)}) \\le k$ for all $t$, in the [[word metric|VCHJVR9NGH]].\n\nExample: $G = \\Z^2 = \\langle a, b \\rangle$ with $A = \\set{a, a^{-1}, b, b^{-1}}$ and $L$ the set of words $a^m b^n$, $m, n \\in \\Z$, which is regular.\nTwo such words naming elements that differ by one generator differ by one in the exponent of $a$ or of $b$.\nTheir prefixes stay at distance at most $2$, so $L$ has the fellow traveller property.\n\nNon-example: the integer Heisenberg group is not automatic.\nIt is finitely presented, but its Dehn function grows like $n^3$, while every automatic group satisfies a quadratic isoperimetric inequality.\n\nBeing automatic does not depend on the choice of the finite generating set.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "J5XQRP1SNQ",
    "kind": "definition",
    "msc": [
      "20F65",
      "20F10",
      "68Q45"
    ],
    "name": "Automatic group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Automatic_group"
      }
    ],
    "tag": "GMPJ88PAQE",
    "uses": [
      "G5S1M9",
      "0RP7VWQ0YN",
      "VCHJVR9NGH"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/GMPJ88PAQE/",
  "links": [
    {
      "field": "defines",
      "target": "J5XQRP1SNQ"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    },
    {
      "field": "uses",
      "target": "0RP7VWQ0YN"
    },
    {
      "field": "uses",
      "target": "VCHJVR9NGH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GMPJ88PAQE.md",
  "msc": [
    "20F65",
    "20F10",
    "68Q45"
  ],
  "name": "Automatic group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GMP/GMPJ88PAQE.md",
  "status": null,
  "tag": "GMPJ88PAQE",
  "uri": "m2://GMPJ88PAQE",
  "url": "https://math2.tamnd.com/wiki/automatic-group-definition/",
  "writing": {
    "tag": "GMPJ88PAQE",
    "kind": "definition",
    "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
      }
    ]
  }
}