Essential Circles and Gromov-Hausdorff Convergence of Covers

TL;DR

This paper explores the use of essential circles in compact geodesic spaces to analyze their homotopy spectra, introduce circle covers, and understand Gromov-Hausdorff convergence, providing new insights into fundamental group generators.

Contribution

It introduces circle covers based on essential circles, demonstrating their stability under Gromov-Hausdorff convergence and providing a new finite generating set for fundamental groups.

Findings

Essential circles determine the homotopy critical spectrum.

Circle covers are Gromov-Hausdorff closed and have finite deck groups.

A new finite generating set for fundamental groups is established.

Abstract

We give various applications of essential circles (introduced in an earlier paper by the authors) in a compact geodesic space X. Essential circles completely determine the homotopy critical spectrum of X, which we show is precisely 2/3 the covering spectrum of Sormani-Wei. We use finite collections of essential circles to define "circle covers," which extend and contain as special cases the delta-covers of Sormani and Wei (equivalently the epsilon-covers of the authors); the constructions are metric adaptations of those utilized by Berestovskii-Plaut in the construction of entourage covers of uniform spaces. We show that, unlike delta- and epsilon-covers, circle covers are in a sense closed with respect to Gromov-Hausdorff convergence, and we prove a finiteness theorem concerning their deck groups that does not hold for covering maps in general. This allows us to completely understand the structure of Gromov-Hausdorff limits of delta-covers. Also, we use essential circles to strengthen a theorem of E. Cartan by finding a new (even for compact Riemannian manifolds) finite set of generators of the fundamental group of a semilocally simply connected compact geodesic space. We conjecture that there is always a generating set of this sort having minimal cardinality among all generating sets.