Discrete homotopies and the fundamental group

Abstract

We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g1, ..., g k of length at most 2D and relators of the form gigm = gj . 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 semi-locally 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.

The revised and uniform fundamental groups and universal covers of geodesic spaces

“…The proof of Theorem 1.1 will mostly show that the other statements are equivalent to 2, but we include 1 for both emphasis and reference. Moreover, the implication 2 ⇒ 3iii follows directly from Lemma 4.1 and a result of Plaut and the author in [21], which is recalled in Section 2. The implications 3iii ⇒ 3ii ⇒ 3i are clear, and likewise for the parts of 4.…”

“…There is a natural metric, d ε , on X ε that makes ϕ ε a regular metric covering map (cf. [11], [21], or [29]). We call X ε and its deck group, π ε (X), the ε-cover and ε-group of X, respectively.…”

Intrinsic flat convergence of covering spaces

“…Can one define new spectra which capture part of the covering spectrum but behave better under intrinsic flat convergence? One possible approach to avoid the disappearance of elements in the covering spectrum would be to consider the work of Plaut and Wilkins [PW13], in which elements of the covering spectra are found using -homotopies. If one requires these homotopies to be built from points with uniform conditions that guarantee the points don't disappear under intrinsic flat convergence, then one may be able to define a new spectra which converge well under intrinsic flat convergence.…”

“…Additional work on the covering spectrum and related notions has been conducted by Bart DeSmit, John Ennis, Ruth Gornet, Conrad Plaut, Craig Sutton, Jay Wilkins and Will Wylie [dSGS10], [dSGS12], [EW06], [PW13], [Wil13], [Wyl06].…”

Various covering spectra for complete metric spaces

“…This is intriguing in light of the work of Colin de Verdiere and Duistermaat-Guillemin relating the length and Laplace spectra of compact Riemannian manifolds [10,15]. A recent extension of the notion of covering spectrum to a larger class of spaces which is called the critical spectrum has been studied by Wilkins, Plaut, Conant, Curnutte, Jones, Pueschel and Walpole [41] [29] [42] [11]. The key definitions, theorems and examples are reviewed in Section 2.…”