Let $(M^{n},g,p)$ satisfy $\Ric_{g}\equiv\lambda g$ and $|\lambda|\leq n-1$. Given a manifold $(Z^{k},z^{k})$ with $k=\dim(Z^{k})<n$, there are uniform constants $\delta_{0}>0$, $w_{0}>0$ and $C_{0}>0$ which depend only on $n$ and the geometry of $B_{1}(z^{k})$ such that the following property holds: if

then the group $\Gamma_{\delta_{0}}(p)\equiv\Image[\pi_{1}(B_{\delta_{0}}(p))\to\pi_{1}(B_{2}(p))]$ has a nilpotent subgroup $\mathcal{N}$ of index bounded by $w_{0}$ such that $\rank(\mathcal{N})\leq n-k$.

Furthermore, if $\rank(\mathcal{N})=n-k$, then $\sup\limits_{B_{1}(p)}|\Rm|\leq C_{0}$. Conversely, if $\sup\limits_{B_{3}(p)}|\Rm|\leq C_{0}$, then $\rank(\mathcal{N})=n-k$.