Polynomial growth of a finitely generated two-step nilpotent group
= Polynomial growth of a finitely generated two-step nilpotent group
Every finitely generated <two-step nilpotent group> has <polynomial growth of a group>. If $s_1,\ldots,s_k$ generate it, centrality of the commutators collects every word of length at most $n$ into
$$
s_1^{a_1}\cdots s_k^{a_k}\prod_{i<j}[s_i,s_j]^{b_{ij}},
$$
where $|a_i|\leq n$ and $|b_{ij}|=O(n^2)$. There are therefore only polynomially many possible collected words.