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  "body": "\n## Summary\n\nAn automatic group is a finitely generated group whose elements can be named by words of a regular language.\nThe naming must allow finite automata to check when two named elements differ by a generator.\nThe idea comes from the work of Cannon, Epstein, Thurston and others on computing with groups.\nAutomatic groups are finitely presented, their word problem can be solved in quadratic time, and their [[Dehn function|Q4HGMRA9JQ]] is at most quadratic.\n[[Hyperbolic groups|JWJ8YPR7EB]], $\\Z^n$, braid groups and many [[Coxeter groups|G6FND7X69J]] are automatic.\n\n## Explanations\n\n### L1 human\n\nA finite automaton is a machine with finitely many states that reads a word one letter at a time.\nIn an automatic group, such a machine can recognise a chosen list of normal-form words, one or more for each element.\nFurther machines, reading two words side by side, can tell whether the elements they name differ by one generator.\nIn $\\Z^2$ with generators $a$ and $b$, the words $a^m b^n$ with $m, n \\in \\Z$ form such a list.\nThis makes many computations in the group fast and mechanical.\n",
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