Hausdorff convergence and universal covers

Authors:
Christina Sormani and Guofang Wei

Journal:
Trans. Amer. Math. Soc. **353** (2001), 3585-3602

MSC (1991):
Primary 53C20

DOI:
https://doi.org/10.1090/S0002-9947-01-02802-1

Published electronically:
April 26, 2001

MathSciNet review:
1837249

Full-text PDF

Abstract | References | Similar Articles | Additional Information

We prove that if is the Gromov-Hausdorff limit of a sequence of compact manifolds, , with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then has a universal cover. We then show that, for sufficiently large, the fundamental group of has a surjective homeomorphism onto the group of deck transforms of . Finally, in the non-collapsed case where the 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 are only assumed to be compact length spaces with a uniform upper bound on diameter.

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Additional Information

**Christina Sormani**

Affiliation:
Department of Mathematics and Computer Science, Lehman College, City University of New York, Bronx, New York 10468

Email:
sormani@g230.lehman.cuny.edu

**Guofang Wei**

Affiliation:
Department of Mathematics, University of California, Santa Barbara, California 93106

Email:
wei@math.ucsb.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02802-1

Received by editor(s):
September 6, 2000

Published electronically:
April 26, 2001

Additional Notes:
Partially supported by NSF Grant #DMS-9971833

Article copyright:
© Copyright 2001
American Mathematical Society