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Homology and closed geodesics in a compact Riemann surface. (English) Zbl 0647.53036
The authors examine the asymptotic distribution of the lengths of prime closed geodesics in a fixed homology class on a compact Riemann surface M of constant negative curvature -1. If P(x,a) \((a\in H_ 1(M,{\mathbb{Z}}))\) is the number of such geodesics of length smaller than x, then they show that P(x,a) behaves like \((g-1)^ g e^ x/x^{g+1},\) where g is the genus of M. The same result was obtained by R. Phillips and P. Sarnak for compact hyperbolic manifolds [Duke Math. J. 55, 287-297 (1987; Zbl 0642.53050)].
Reviewer: M.Burger

53C22 Geodesics in global differential geometry
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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