Hausdorff convergence and universal covers

Abstract

We prove that if Y is the Gromov-Hausdorff limit of a sequence of compact manifolds, Min, with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then Y has a universal cover. We then show that, for i sufficiently large, the fundamental group of Mi has a surjective homeomorphism onto the group of deck transforms of Y . Finally, in the non-collapsed case where the Mi have an additional uniform lower bound on volume, we prove that the kernels of these surjective maps are finite with a uniform bound on their cardinality. A number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the Mi are only assumed to be compact length spaces with a uniform upper bound on diameter.