#### Volume 14, issue 4 (2010)

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Describing the universal cover of a noncompact limit

### John Ennis and Guofang Wei

Geometry & Topology 14 (2010) 2479–2496
##### Abstract

Suppose that $X$ is the Gromov–Hausdorff limit of a sequence of Riemannian manifolds ${M}_{i}^{n}$ with a uniform lower bound on Ricci curvature. In a previous paper the authors showed that when $X$ is compact the universal cover $\stackrel{̃}{X}$ is a quotient of the Gromov–Hausdorff limit of the universal covers ${\stackrel{̃}{M}}_{i}^{n}$. This is not true when $X$ is noncompact. In this paper we introduce the notion of pseudo-nullhomotopic loops and give a description of the universal cover of a noncompact limit space in terms of the covering spaces of balls of increasing size in the sequence.

##### Keywords
universal cover, Gromov–Hausdorff limit
Primary: 53C20
##### Publication
Received: 2 August 2007
Revised: 6 May 2010
Accepted: 17 October 2010
Published: 10 December 2010
Proposed: Gang Tian
Seconded: Tobias H Colding, Steven Ferry
##### Authors
 John Ennis The Institute for Perception 7629 Hull Street Rd Richmond VA 23235 USA Guofang Wei Mathematics Department University of California, Santa Barbara Santa Barbara CA 93106 USA