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Geodesics in homology classes. (English) Zbl 0642.53050
Let \(M\) be a hyperbolic manifold, and \(\pi_{\beta}(x)\) the number of prime closed geodesics of length at most \(x\) in a given homology class \(H^ 1(M, \mathbb Z)\). Motivated by work of G. A. Margulis [Funct. Anal. Appl. 3, 335–336 (1970); translation from Funkts. Anal. Prilozh. 3, No. 4, 89–90 (1969; Zbl 0207.20305)] and of Adachi and Sunada, the author investigates the asymptotic behaviour of \(\pi_{\beta}(x)\). In fact given a surjective homomorphism \(\psi: \pi_ 1(M)\to \Lambda\) of the fundamental group onto an abelian group, and \(\beta\in \Lambda\), an asymptotic expansion for \(\pi_{\beta}(x)\) is obtained. This expansion involves the rank of \(\Lambda\) as well as the order of the torsion of \(\Lambda\).

53C22 Geodesics in global differential geometry
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
Full Text: DOI
[1] T. Adachi and T. Sunada, Homology of closed geodesics in a negatively curved manifold , preprint, 1986. · Zbl 0618.58028
[2] C. Epstein, The spectral theory of geometrically periodic hyperbolic \(3\)-manifolds , Mem. Amer. Math. Soc. 58 (1985), no. 335, ix+161. · Zbl 0584.58047
[3] H. Farkas and I. Kra, Riemann Surfaces , Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1980. · Zbl 0475.30001
[4] G. A. Margulis, Applications of ergodic theory to the investigations of manifolds of negative curvature , Func. Anal. and Appl. 3 (1969), 335-336. · Zbl 0207.20305 · doi:10.1007/BF01076325
[5] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series , J. Indian Math. Soc. (N.S.) 20 (1956), 47-87. · Zbl 0072.08201
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