  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.
