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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 $$\Lambda$$.

MSC:
 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
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References:
 [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 [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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