Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set $K subset X$ the sectional curvatures of $g$ are identically equal to minus one. We prove that the c
ounting function for the resolvent resonances has maximal order of growth $(n+1)$ generically for such manifolds.
We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three
. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.
