  For quotients of the $n+1$-dimensional hyperbolic space by a convex
co-compact group $\Gamma$, we obtain a formula relating the renormalized trace
of the wave operator with the resonances of the Laplacian and some conformal
invariants of the boundary, generalizing a formula of Guillop\'e and Zworski in
dimension 2. By writing this trace with the length spectrum, we prove precise
asymptotics of the number of closed geodesics with an effective, exponentially
small error term when the dimension of the limit set of $\Gamma$ is greater
than $n/2$.
