Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 3 / c

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 $s_1,\dots ,s_k$. 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 formThe generator exponents satisfy $|a_i|\le n$. At most $n(n-1)/2$ exchanges are needed during collection, so $|b_{ij}|\le n^2$ is a sufficient bound. With $m=k(k-1)/2$, the number of possible collected expressions is at mostRelations can only reduce this count. This proves the polynomial growth of a finitely generated two-step nilpotent group, so every finitely generated two-step nilpotent group has polynomial growth.