Let $G$ be an infinite finitely generated torsion-free nilpotent group. Let $H$ be a finite-index subgroup of index $n$. Let $G^\prime$, $H^\prime$ denote the derived groups of $G$ and $H$ respectively.
I want to show that the index $|G^\prime \cap H : H^\prime|$ is bounded by a polynomial in $n = |G:H|$.
I suggest the polynomial $n^{h(G)}$, where $h(G)$ denotes the Hirsch length of $G$ should do the trick, perhaps even $n^{h(G^\prime)}$, but alas I am unable to prove such!