Weak stability of the geodesic flow and Preissmann's theorem

Let (M, g) be a compact, differentiable Riemannian manifold without conjuga te points and bounded asymptote. We show that, if the geodesic flow of (M, g) is either topologically stable, or satisfies the epsilon-shadowing prope rty for some appropriate epsilon > 0, then every abelian subgroup of the fu ndamental group of M is infinite cyclic. The proof is based on the existenc e of homoclinic geodesics in perturbations of (M, g), whenever there is a s ubgroup of the fundamental group of M isomorphic to Z x Z.