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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a finitely generated [[associative algebra|3CTW1P4HBP]] over $k$ with identity.\nLet $V \\subseteq A$ be a finite-dimensional [[subspace|000AYPA9AJ]] that contains $1$ and generates $A$ as an algebra.\nFor $n \\ge 0$, let $V^n$ be the span of all products $v_1 v_2 \\cdots v_n$ with $v_i \\in V$, with $V^0 = k$.\nThe *growth function* of $A$ with respect to $V$ is\n$$d_V(n) = \\dim_k V^n.$$\nFor functions $f, g \\colon \\N \\to \\N$, write $f \\preceq g$ if there is an integer $c \\ge 1$ with $f(n) \\le g(cn)$ for all $n$, and $f \\sim g$ if $f \\preceq g$ and $g \\preceq f$.\nThe *growth* of $A$ is the class of $d_V$ under $\\sim$.\n\n## Remarks\n\nThe growth does not depend on $V$.\nIf $W$ is another such subspace, then $W \\subseteq V^c$ for some $c$, because $W$ is finite-dimensional and $\\bigcup_n V^n = A$; so $d_W(n) \\le d_V(cn)$, and symmetrically.\n\nThe algebra $A$ has *polynomial growth* if $d_V(n) \\le C n^r$ for some constants $C, r$.\nIt has *exponential growth* if $d_V(n) \\ge a^n$ for some $a \u003e 1$ and all $n$.\nOtherwise it has *intermediate growth*.\n\nExample: for $A = k[x, y]$ and $V = k + kx + ky$, the space $V^n$ has the basis of monomials $x^i y^j$ with $i + j \\le n$.\nSo $d_V(n) = (n+1)(n+2)/2$, and $A$ has polynomial growth.\n\nNon-example of the setting: the field $k(x)$ of rational functions is not a finitely generated $k$-algebra, so no growth function is defined for it in this way.\n\nFor a finitely generated [[group|4CMRM3975W]] $G$ with a finite symmetric generating set $S$, take $A = k[G]$, the [[group algebra|MNP2HBWB5Q]], and $V$ the span of $\\set{1} \\cup S$.\nThen $V^n$ has the basis of group elements of word length at most $n$, so the growth of $k[G]$ is the growth of $G$.\n",
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