Hausdorff Convergence and Universal Covers

Details Hausdorff Convergence and Universal Covers Hausdorff Convergence and Universal Covers

2000-08-29

arxiv.org/abs/math/0008218v1

Tran. Amer. Math. Soci. 353 (2001) 3585-3602

Mathematics

Differential Geometry

17 Pages

Scientific paper

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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