Convergence of Ricci-limit spaces under bounded Ricci curvature and local covering geometry I

We extend Cheeger-Gromov's and Anderson's convergence theorems to regular limit spaces of manifolds with bounded Ricci curvature and local covering geometry, by establishing the \(C^{1,\alpha}\)-regularities that are the best one may expect on those Ricci-limit spaces. As an application we prove an optimal generalization of Fukaya's fibration theorem on collapsed manifolds with bounded Ricci curvature, which also improves the original version to \(C^{1,\alpha}\) limit spaces.