Geodesics in homology classes.

*(English)*Zbl 0642.53050Let \(M\) 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 \(\Lambda\).

Reviewer: Karsten Grove (College Park)

##### 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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\textit{R. Phillips} and \textit{P. Sarnak}, Duke Math. J. 55, 287--297 (1987; Zbl 0642.53050)

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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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