A beautiful theorem of Brooks says that, for a wide class of Riemannian manifolds, the bottom of the spectrum of the Laplacian on a regular cover is equal to the bottom of the spectrum of the base if and only if the covering group is amenable. In the case where the base manifold is a quotient of a simply connected manifold with pinched negative curvatures by a convex co-compact group, we will give a analogous results for critical exponents and for the growth of closed geodesics. This is joint work with Rhiannon Dougall.