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Describing the Universal Cover of a Compact Limit John Ennis Guofang Wei

Summary: Describing the Universal Cover of a Compact Limit
John Ennis Guofang Wei
If X is the Gromov-Hausdorff limit of a sequence of Riemannian manifolds
i with a uniform lower bound on Ricci curvature, Sormani and Wei have
shown that the universal cover ~X of X exists [13, 14]. 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.
1 Introduction
In 1981 Gromov proved that any finitely generated group has polynomial growth if
and only if it is almost nilpotent [7]. In his proof, Gromov introduced the Gromov-
Hausdorff distance between metric spaces [7, 8, 9]. This distance has proven to
be especially useful in the study of n-dimensional manifolds with Ricci curvature
uniformly bounded below since any sequence of such manifolds has a convergent
subsequence [10]. Hence we can follow an approach familiar to analysts, and consider
the closure of the class of all such manifolds. The limit spaces of this class have
path metrics, and one can study these limit spaces from a geometric or topological


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics