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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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