Describing the universal cover of a compact limit

If X is the Gromov-Hausdorff limit of a sequence of Riemannian manifolds Min with a uniform lower bound on Ricci curvature, Sormani and Wei have shown that the universal cover X˜ of X exists [C. Sormani, G. Wei, Hausdorff convergence and universal covers, Trans. Amer. Math. Soc. 353 (9) (2001) 3585-3602 (electronic)]; [C. Sormani, G. Wei, Universal covers for Hausdorff limits of noncompact spaces, Trans. Amer. Math. Soc. 356 (3) (2004) 1233-1270 (electronic). [15]]. For the case where X is compact, we provide a description of X˜ in terms of the universal covers M˜i of the manifolds. More specifically we show that if X¯ is the pointed Gromov-Hausdorff limit of the universal covers M˜i then there is a subgroup H of Iso(X¯) such that X˜=X¯/H. We call H the small action limit group and prove a similar result for compact length spaces with uniformly bounded dimension.