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  "body": "\n## Summary\n\nThe growth function of a [[finitely generated group|082DWWXMRJ]] counts how many elements can be written as words of length at most $n$ in the generators.\nIts rate of increase, up to a natural equivalence, does not depend on the generating set, and it is a [[quasi-isometry|6FKYSH729T]] invariant.\nGroups split into groups of polynomial growth, of exponential growth, and of intermediate growth.\n[[Gromov's theorem|RSBFHHPTWX]] identifies the groups of polynomial growth as the virtually nilpotent groups.\n\n## Explanations\n\n### L1 human\n\nStart at the identity of the group and take $n$ steps, each step a multiplication by a generator or its inverse.\nCount how many different elements you can reach.\nIn $\\Z$ with generator $1$, after at most $n$ steps you reach the $2n + 1$ integers from $-n$ to $n$, which grows like a line.\nIn $\\Z^2$ you reach a diamond of about $2n^2$ points, which grows like an area.\nIn a free group on two generators, the number of reachable elements roughly triples with each step, which is exponential growth.\n",
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