Describing the universal cover of a noncompact limit

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 $\tilde{X}$ is a quotient of the Gromov–Hausdorff limit of the universal covers ${\tilde{M}}_i^n$. 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.