Wave 0Trace and length spectrum on convex cocompact hyperbolic manifolds
Abstract
For quotients of the $n+1$dimensional hyperbolic space by a convex cocompact 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 $n/2$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606223
 Bibcode:
 2006math......6223G
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Spectral Theory;
 58J50;
 11F72;
 35P25
 EPrint:
 13 pages. One typo corrected in the formula of S(s) in proposition 2.2. Already fixed in the published version