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

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

2006-06-09

arxiv.org/abs/math/0606223v1

Mathematics

Differential Geometry

13 pages

Scientific paper

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 $n/2$.

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