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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hausdorff convergence and universal covers


Authors: Christina Sormani and Guofang Wei
Journal: Trans. Amer. Math. Soc. 353 (2001), 3585-3602
MSC (1991): Primary 53C20
Published electronically: April 26, 2001
MathSciNet review: 1837249
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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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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: http://dx.doi.org/10.1090/S0002-9947-01-02802-1
PII: S 0002-9947(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