Various Covering Spectra for Complete Metric Spaces

TL;DR

This paper explores various covering spectra in complete noncompact length spaces, relating them to the shift spectrum, and introduces rescaled concepts to analyze fundamental groups, with implications for Riemannian manifolds with Ricci curvature bounds.

Contribution

It introduces rescaled covering spectra and slipping groups, establishing finiteness of fundamental groups for certain Riemannian manifolds with Ricci curvature bounds.

Findings

Covering spectrum relates to the shift spectrum in length spaces.

Rescaled covering spectrum provides new insights into fundamental groups.

Certain manifolds with nonnegative Ricci curvature have finite fundamental groups.

Abstract

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.