Wave 0-trace and Length Spectrum on Convex Co-compact Hyperbolic Manifolds

For convex co-compact hyperbolic quotients $\Gamma/\Bbb{H}^{n+1}$, we obtain aformula relating the 0-trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0-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 $n/2$.

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Published 1 January 2006