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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

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
Full Text: DOI
[1] C. Epstein, The spectral theory of geometrically periodic hyperbolic \(3\)-manifolds , Mem. Amer. Math. Soc. 58 (1985), no. 335, ix+161. · Zbl 0584.58047
[2] D. Hejhal, The Selberg trace formula for \(\mathrm PSl(2,\mathbf R)\), vol. 2 , Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983. · Zbl 0543.10020
[3] T. Kato, Perturbation Theory for Linear Operators , 2d ed., Springer-Verlag, New York, 1976. · Zbl 0342.47009
[4] T. Adachi and T. Sunada, Homology and closed geodesics in a compact Riemann surface , preprint, 1986.
[5] P. Lax and R. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces , J. Funct. Anal. 46 (1982), no. 3, 280-350. · Zbl 0497.30036 · doi:10.1016/0022-1236(82)90050-7
[6] R. Phillips and P. Sarnak, Geodesics in homology classes , preprint, 1986. · Zbl 0642.53050 · doi:10.1215/S0012-7094-87-05515-3
[7] R. S. Phillips and P. Sarnak, The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups , Acta Math. 155 (1985), no. 3-4, 173-241. · Zbl 0611.30037 · doi:10.1007/BF02392542
[8] D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics , Acta Math. 149 (1982), no. 3-4, 215-237. · Zbl 0517.58028 · doi:10.1007/BF02392354
[9] G. Watson, The Theory of Bessel Functions , Macmillan, New York, 1948. · Zbl 0038.42507
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