= Solution
A finitely generated group has <polynomial growth of a group> if its growth function is bounded above by $C(1+r)^D$ for some constants $C,D$.
Let $G$ be generated by $s_1,\ldots,s_k$. Since $G$ is a <two-step nilpotent group>, every <group commutator> $[s_i,s_j]$ is central. Commuting letters past one another therefore collects every word of length at most $n$ into the form
$$
s_1^{a_1}\cdots s_k^{a_k}
\prod_{1\leq i<j\leq k}[s_i,s_j]^{b_{ij}}.
$$
The generator exponents satisfy $|a_i|\leq n$. At most $n(n-1)/2$ exchanges are needed during collection, so $|b_{ij}|\leq n^2$ is a sufficient bound. With $m=k(k-1)/2$, the number of possible collected expressions is at most
$$
(2n+1)^k(2n^2+1)^m=O\!\left(n^{k+2m}\right).
$$
Relations can only reduce this count. This proves the <polynomial growth of a finitely generated two-step nilpotent group>, so \b[every finitely generated two-step nilpotent group has polynomial growth].
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