  We study various covering spectra for complete noncompact length spaces with
universal covers (including Riemannian manifolds and the pointed Gromov
Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci
curvature). We relate the covering spectrum to the (marked) shift spectrum of
such a space. We define the slipping group generated by elements of the
fundamental group whose translative lengths are 0. We introduce a rescaled
length, the rescaled covering spectrum and the rescaled slipping group.
Applying these notions we prove that certain complete noncompact Riemannian
manifolds with nonnegative or positive Ricci curvature have finite fundamental
groups. Throughout we suggest further problems both for those interested in
Riemannian geometry and those interested in metric space theory.
