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 Guillopé 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 $n/2$.
arXiv Mathematics e-prints
- Pub Date:
- June 2006
- Mathematics - Differential Geometry;
- Mathematics - Spectral Theory;
- 13 pages. One typo corrected in the formula of S(s) in proposition 2.2. Already fixed in the published version