If X is the Gromov–Hausdorff limit of a sequence of Riemannian manifolds Mni 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). ]. For the case where X is compact, we provide a description of ˜X in terms of the universal covers ˜Mi of the manifolds. More specifically we show that if ¯X is the pointed Gromov–Hausdorff limit of the universal covers ˜Mi 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.
This article appears as:
Ennis, J. and Wei, G. (2006). Describing the universal cover of a compact limit. Differential Geometry and its Applications, 24(5), 554–562.
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Describing the universal cover of a compact limit
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