= Finite fundamental group from a uniform positive Ricci bound
{title2=$\operatorname{Ric}\ge(n-1)\kappa g\ \Longrightarrow\ |\pi_1(M)|<\infty$}
If a connected complete <Riemannian manifold> has $\operatorname{Ric}\ge(n-1)\kappa g$ with $\kappa>0$, its <universal cover> with the lifted metric is complete and satisfies the same bound. The <Bonnet-Myers theorem> makes that cover compact. A covering fiber is closed and discrete and hence finite; the number of its elements equals the order of the <fundamental group>. Applying the diameter theorem only to the base would not prove this conclusion: the positive lower bound must also be used on the cover.
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