On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups

We prove that a circle bundle over a closed oriented aspherical manifold with hy-perbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known by the work of Brooks and Goldman on the Seifert volume. As a consequence, we verify the following strong version of a problem of Hopf for the above class of manifolds: every self-map of nonzero degree of a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group is either homotopic to a homeomorphism or homotopic to a nontrivial covering and the bundle is virtually trivial. As another application, we derive the first examples of nonvanishing numerical invariants that are monotone with respect to the mapping degree on nontrivial circle bundles over aspherical manifolds with hyperbolic fundamental groups in any dimension.