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  "body": "\n## Description\n\nLet $\\set{0, 1}^*$ be the set of finite words in the letters $0$ and $1$, the vertices of the infinite rooted binary [[tree|062D9KXJB1]].\nDefine four maps $a, b, c, d$ of $\\set{0, 1}^*$ to itself by recursion on the length of a word $w$.\nEach map sends the empty word to the empty word, and:\n$$a(0w) = 1w, \\quad a(1w) = 0w,$$\n$$b(0w) = 0\\,a(w), \\quad b(1w) = 1\\,c(w),$$\n$$c(0w) = 0\\,a(w), \\quad c(1w) = 1\\,d(w),$$\n$$d(0w) = 0w, \\quad d(1w) = 1\\,b(w).$$\nEach map is a bijection that preserves the length of words and the tree structure.\nThe *Grigorchuk group* $\\Gamma$ is the [[group|G5S1M9]] generated by $a, b, c, d$ under composition.\n\n## Remarks\n\nEach generator has order $2$, and $\\set{1, b, c, d}$ is a group isomorphic to $\\Z/2 \\times \\Z/2$ with $bc = d$.\n$\\Gamma$ is infinite, finitely generated, and every element has order a power of $2$, so it answers the general [[Burnside problem|XWEMJ7T2XV]] in the negative.\nIt is [[residually finite|HEE922EHQ8]]: the action on the words of length $n$ gives finite quotients that separate all elements.\nIt has [[growth|4CMRM3975W]] faster than every polynomial and slower than every exponential; it was the first group known with this intermediate growth.\nIt is [[amenable|CEAT9BESKK]] and not finitely presented.\n\n## Explanations\n\n### L1 human\n\nThe Grigorchuk group is a group of symmetries of an infinite binary tree, generated by four explicit moves.\nThe move $a$ swaps the two halves of the tree.\nThe moves $b$, $c$ and $d$ act on the two halves separately, each time by $a$ or by another of $b$, $c$, $d$, in a repeating pattern.\nFrom these simple rules comes a group with properties that were long thought hard to find together.\nIt is infinite, every element repeated enough times gives the identity, and its size grows at an unusual speed.\n",
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