Universal covers for Hausdorff limits of noncompact spaces

Authors:
Christina Sormani and Guofang Wei

Journal:
Trans. Amer. Math. Soc. **356** (2004), 1233-1270

MSC (2000):
Primary 53C20

Published electronically:
October 6, 2003

MathSciNet review:
2021619

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if is the Gromov-Hausdorff limit of a sequence of complete manifolds, , with a uniform lower bound on Ricci curvature, then has a universal cover.

**[AbGl]**Uwe Abresch and Detlef Gromoll,*On complete manifolds with nonnegative Ricci curvature*, J. Amer. Math. Soc.**3**(1990), no. 2, 355–374. MR**1030656**, 10.1090/S0894-0347-1990-1030656-6**[An1]**Michael T. Anderson,*Convergence and rigidity of manifolds under Ricci curvature bounds*, Invent. Math.**102**(1990), no. 2, 429–445. MR**1074481**, 10.1007/BF01233434**[An2]**Michael T. Anderson,*On the topology of complete manifolds of nonnegative Ricci curvature*, Topology**29**(1990), no. 1, 41–55. MR**1046624**, 10.1016/0040-9383(90)90024-E**[BiCr]**Richard L. Bishop and Richard J. Crittenden,*Geometry of manifolds*, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1964 original. MR**1852066****[BoMe]**B. H. Bowditch and G. Mess,*A 4-dimensional Kleinian group*, Trans. Amer. Math. Soc.**344**(1994), no. 1, 391–405. MR**1240944**, 10.1090/S0002-9947-1994-1240944-6**[BBI]**Dmitri Burago, Yuri Burago, and Sergei Ivanov,*A course in metric geometry*, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR**1835418****[ChCo1]**Jeff Cheeger and Tobias H. Colding,*Lower bounds on Ricci curvature and the almost rigidity of warped products*, Ann. of Math. (2)**144**(1996), no. 1, 189–237. MR**1405949**, 10.2307/2118589**[ChCo2]**Jeff Cheeger and Tobias H. Colding,*On the structure of spaces with Ricci curvature bounded below. I*, J. Differential Geom.**46**(1997), no. 3, 406–480. MR**1484888****[ChCo3]**Jeff Cheeger and Tobias H. Colding,*On the structure of spaces with Ricci curvature bounded below. II*, J. Differential Geom.**54**(2000), no. 1, 13–35. MR**1815410****[ChCo4]**Jeff Cheeger and Tobias H. Colding,*On the structure of spaces with Ricci curvature bounded below. III*, J. Differential Geom.**54**(2000), no. 1, 37–74. MR**1815411****[Fed]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[Gr]**Misha Gromov,*Metric structures for Riemannian and non-Riemannian spaces*, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR**1699320****[GP]**Karsten Grove and Peter Petersen,*Manifolds near the boundary of existence*, J. Differential Geom.**33**(1991), no. 2, 379–394. MR**1094462****[Ma]**William S. Massey,*A basic course in algebraic topology*, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991. MR**1095046****[Me]**X. Menguy,*Examples with bounded diameter growth and infinite topological type*, Duke Math. J.**102**(2000), no. 3, 403–412. MR**1756103**, 10.1215/S0012-7094-00-10232-3**[Mi]**J. Milnor,*A note on curvature and fundamental group*, J. Differential Geometry**2**(1968), 1–7. MR**0232311****[Mun]**M. E. Munroe,*Introduction to measure and integration*, Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1953. MR**0053186****[Nab]**Philippe Nabonnand,*Sur les variétés riemanniennes complètes à courbure de Ricci positive*, C. R. Acad. Sci. Paris Sér. A-B**291**(1980), no. 10, A591–A593 (French, with English summary). MR**600003****[Pl1]**G. Ya. Perel′man,*Elements of Morse theory on Aleksandrov spaces*, Algebra i Analiz**5**(1993), no. 1, 232–241 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**5**(1994), no. 1, 205–213. MR**1220498****[Pl2]**G. Perelman,*Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers*, Comparison geometry (Berkeley, CA, 1993–94) Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 157–163. MR**1452872****[Pe]**Peter Petersen,*Riemannian geometry*, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 1998. MR**1480173****[Po]**L. Potyagaĭlo,*Finitely generated Kleinian groups in 3-space and 3-manifolds of infinite homotopy type*, Trans. Amer. Math. Soc.**344**(1994), no. 1, 57–77. MR**1250823**, 10.1090/S0002-9947-1994-1250823-6**[So1]**Christina Sormani,*Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups*, J. Differential Geom.**54**(2000), no. 3, 547–559. MR**1823314****[So2]**C. Sormani,*On loops representing elements of the fundamental group of a complete manifold with nonnegative Ricci curvature*, Indiana Univ. Math. J.**50**(2001), no. 4, 1867–1883. MR**1889085**, 10.1512/iumj.2001.50.2048**[SoWei]**Christina Sormani and Guofang Wei,*Hausdorff convergence and universal covers*, Trans. Amer. Math. Soc.**353**(2001), no. 9, 3585–3602 (electronic). MR**1837249**, 10.1090/S0002-9947-01-02802-1**[Sp]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****[Wei]**Guofang Wei,*Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups*, Bull. Amer. Math. Soc. (N.S.)**19**(1988), no. 1, 311–313. MR**940494**, 10.1090/S0273-0979-1988-15653-4

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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/S0002-9947-03-03412-3

Received by editor(s):
July 24, 2002

Received by editor(s) in revised form:
February 28, 2003

Published electronically:
October 6, 2003

Additional Notes:
The first author was partially supported by NSF Grant # DMS-0102279 and a grant from The City University of New York PSC-CUNY Research Award Program

The second author was partially supported by NSF Grant # DMS-9971833

Article copyright:
© Copyright 2003
American Mathematical Society