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Asymptotics for closed geodesics in a homology class, the finite volume case. (English) Zbl 0648.58041
The author considers the problem of counting the number of closed geodesics on a hyperbolic manifold M which are of length $$\leq \lambda$$ and which represent a given class of $$H_ 1(M,{\mathbb{Z}})$$. The author finds an asymptotic formula for this for large $$\lambda$$ which, incidently, does not depend on the homology class in question. The nature of this formula depends on the dimension of M and on the number of cusps (M is assumed to be complete and of finite volume). The technique involves perturbation theory applied to the spectral theory of 1- dimensional flat bundles over M and an argument from Fourier analysis analogous to that used in the Hardy-Littlewood method.
Reviewer: S.J.Patterson

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 53C22 Geodesics in global differential geometry 11F99 Discontinuous groups and automorphic forms
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