Pinching estimates for negatively curved manifolds with nilpotent fundamental groups

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 $ϵ>0$, the manifold $M$ admits a complete Riemannian metric of sectional curvature within $[-(k+ϵ{)}^2,-1]$.