Hausdorff convergence and universal covers

Sormani, Christina; Wei, Guofang

doi:10.1090/S0002-9947-01-02802-1

Hausdorff convergence and universal covers
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by Christina Sormani and Guofang Wei PDF

Trans. Amer. Math. Soc. 353 (2001), 3585-3602 Request permission

Abstract:

We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of compact manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature and a uniform 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 homeomorphism onto the group of deck transforms of $Y$. Finally, in the non-collapsed 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 uniform bound on their cardinality. A number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the $M_i$ are only assumed to be compact length spaces with a uniform upper bound on diameter.

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