Closed geodesics on compact orbifolds and on noncompact manifolds

We prove the existence of a nonconstant closed geodesic on every odd-dimensional compact Riemannian orbifold and on every connected, noncontractible Riemannian manifold that admits a cocompact, isometric group action. We deduce new restrictions on a compact Riemannian orbifold without such a geodesic; in particular, its orbifold fundamental group is a rational Poincar'e duality group.