  Let $M$ be a complete Riemannian metric of sectional curvature within
$[-a^2,-1]$ whose fundamental group contains a $k$-step nilpotent subgroup of
finite index. We prove that $a\ge k$ answering a question of M. Gromov.
Furthermore, we show that for any $\epsilon>0$, the manifold $M$ admits a
complete Riemannian metric of sectional curvature within
$[-(k+\epsilon)^2,-1]$.
