Wave 0-Trace and length spectrum on convex co-compact hyperbolic manifolds

Abstract

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 $n/2$.

13 pages. One typo corrected in the formula of S(s) in proposition 2.2. Already fixed in the published version

References in corpus (1)

• Fuchsian groups of the second kind and representations carried by the limit set