Ricci Curvature, Diameter and Fundamental groups

In this note we discuss the fundamental groups and diameters of positively Ricci curved $n$-manifolds. We use a method combining the results about equivarient Hausdorff convergence developed by Fukaya and Yamaguchi with the Ricci version of splitting theorem by Cheeger and Colding to give new information on the topology of compact manifolds with positive Ricci curvature. Moreover, we also obtain a weak Margulis's lemma for manifolds under a lower Ricci curvature bound.