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  "body": "\n## Definition\n\nLet $G$ be a [[finitely generated group|082DWWXMRJ]] and let $S$ be a finite [[generating set|0RP7VWQ0YN]] of $G$.\nThe *word length* $\\abs{g}_S$ of $g \\in G$ is the least $n$ such that $g = s_1 \\cdots s_n$ with each $s_i \\in S \\cup S^{-1}$.\nThe *growth function* of $G$ with respect to $S$ is\n$$\\gamma_S(n) = \\abs{\\set{g \\in G : \\abs{g}_S \\le n}}, \\quad n \\in \\N.$$\nFor functions $f, h \\colon \\N \\to \\N$, write $f \\preceq h$ if there is a constant $C \\ge 1$ with $f(n) \\le h(Cn)$ for all $n$, and $f \\sim h$ if $f \\preceq h$ and $h \\preceq f$.\nThe *growth rate* of $G$ is the $\\sim$-class of $\\gamma_S$.\n\n1. $G$ has *polynomial growth* if $\\gamma_S(n) \\le C n^d$ for some constants $C, d \u003e 0$ and all $n \\ge 1$.\n2. $G$ has *exponential growth* if $\\gamma_S(n) \\ge a^n$ for some $a \u003e 1$ and all $n$.\n3. $G$ has *intermediate growth* if it has neither polynomial nor exponential growth.\n\n## Remarks\n\nThe class of $\\gamma_S$ does not depend on $S$.\nIf $S'$ is another finite generating set, each element of $S'$ is a word of length at most some $C$ in $S$, so $\\gamma_{S'}(n) \\le \\gamma_S(Cn)$.\nHence the three types above do not depend on $S$.\n\nExample: $G = \\Z^2$ with $S = \\set{(1, 0), (0, 1)}$.\nThe elements of length at most $n$ are the $(a, b)$ with $\\abs{a} + \\abs{b} \\le n$, so $\\gamma_S(n) = 2n^2 + 2n + 1$, and $\\Z^2$ has polynomial growth.\n\nExample: the [[free group|0KTW26C80J]] on $\\set{a, b}$.\nThere are $4 \\cdot 3^{k-1}$ reduced words of length $k \\ge 1$, so $\\gamma_S(n) = 2 \\cdot 3^n - 1$, and the growth is exponential.\n\nNon-example: the additive group $\\Q$ is not finitely generated, so the definition does not apply; with an infinite generating set the balls may be infinite.\n\nSince $\\gamma_S(m + n) \\le \\gamma_S(m) \\gamma_S(n)$, the limit $\\lim_{n \\to \\infty} \\gamma_S(n)^{1/n}$ exists, and $G$ has exponential growth exactly when this limit is greater than $1$.\n",
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