Hausdorff convergence and universal covers

We prove that if Y is the Gromov-Hausdorff limit of a sequence of compact m anifolds, M-i(n), with a uniform lower bound on Ricci curvature and a unifo rm upper bound on diameter, then Y has a universal cover. We then show that , for i sufficiently large, the fundamental group of M-i has a surjective h omeomorphism onto the group of deck transforms of Y. Finally, in the non-co llapsed case where the M-i have an additional uniform lower bound on volume , we prove that the kernels of these surjective maps are finite with a unif orm bound on their cardinality. A number of theorems are also proven concer ning the limits of covering spaces and their deck transforms when the M-i a re only assumed to be compact length spaces with a uniform upper bound on d iameter.