Discrete homotopies and the fundamental group

We generalize and strengthen the theorem of Gromov that the fundamental group of any compact Riemannian manifold of diameter at most DD has a set of generators g1,…,gkg1,…,gk of length at most 2D2D and relators of the form gigm=gjgigm=gj. In particular, we obtain an explicit bound for the number kk of generators in terms of the number of “short loops” at every point and the number of balls required to cover a given semi-locally simply connected geodesic space. As a corollary we obtain a fundamental group finiteness theorem (new even for Riemannian manifolds) that replaces the curvature and volume conditions of Anderson and the 1-systole bound of Shen–Wei, by more general geometric hypothesis implied by these conditions. This theorem, in turn, is a special case of a theorem for arbitrary compact geodesic spaces, proved using the method of discrete homotopies introduced by the first author and V. N. Berestovskii. Central to the proof is the notion of “homotopy critical spectrum”, introduced in this paper as a natural consequence of discrete homotopy methods. This spectrum is closely related to the Sormani–Wei covering spectrum which is a subset of the classical length spectrum studied by de Verdiere and Duistermaat–Guillemin. It is completely determined (including multiplicity) by special closed geodesics called “essential circles”.