Fundamental group and contractible closed geodesics

Abstract

We prove the existence of a nonempty class of finitely presented groups with the following property: If the fundamental group of a compact Riemannian manifold M belongs to this class, then there exists a constant c(M) > 1 such that for any sufficiently large x the number of contractible closed geodesics on M of length not exceeding x is greater than c(M)x. In order to prove this result, we give a lower bound for the number of contractible closed geodesics of length ≤ x on a compact Riemannian manifold M in terms of the resource-bounded Kolmogorov complexity of the word problem for π1 (M), thus answering a question posed by Gromov. © 1996 John Wiley & Sons, Inc.