Discrete homotopies and the fundamental group

TL;DR

This paper extends Gromov's theorem to provide explicit bounds on the fundamental group's generators in semilocally simply connected spaces without curvature restrictions, using discrete homotopies and the homotopy critical spectrum.

Contribution

It introduces a new explicit bound for the number of generators of the fundamental group based on short loops and covering properties, generalizing previous results without curvature assumptions.

Findings

Established a bound on the number of fundamental group generators.

Proved a fundamental group finiteness theorem for general geodesic spaces.

Simplified proofs of results related to the covering spectrum.

Abstract

We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g_1,...,g_k of length at most 2D and relators of the form g_ig_m = g_j . In particular, we obtain an explicit bound for the number k of generators in terms of the number "short loops" at every point and the number of balls required to cover a given semilocally simply connected geodesic space. As a consequence we obtain a fundamental group finiteness theorem (new even for Riemannian manifolds) that implies the fundamental group finiteness theorems of Anderson and Shen-Wei. Our theorem requires no curvature bounds, nor lower bounds on volume or 1-systole. We use the method of discrete homotopies introduced by the first author and V. N. Berestovskii. Central to the proof is the notion of the homotopy critical spectrum that is closely related to the covering and length spectra. Discrete methods also allow us to strengthen and simplify the proofs of some results of Sormani-Wei about the covering spectrum.