Hausdorff convergence and universal covers

By C. Sormani, Guofang Wei. 2000

84 citations recorded by the source corpus.

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 Mi has a surjective homeomorphism onto the group of deck transforms of Y . Finally, in the non-collapsed case where the Mi 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 Mi are only assumed to be compact length spaces with a uniform upper bound on diameter.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ae13f6977b25c722a6337ca7dbeebac30996906a

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Universal covers for Hausdorff limits of noncompact spaces (2002, methodology): “We then proved that for a compact length space that has a universal cover, the universal cover is always some δ-cover [ SoWei , Proposition 3.2].”