On pairs of closed geodesics on hyperbolic surfaces
Annales de l'Institut Fourier, Volume 49 (1999) no. 1, p. 1-25

In this article we prove a trace formula for double sums over totally hyperbolic Fuchsian groups Γ. This links the intersection angles and common perpendiculars of pairs of closed geodesics on Γ/H with the inner products of squares of absolute values of eigenfunctions of the hyperbolic laplacian Δ. We then extract quantitative results about the intersection angles and common perpendiculars of these geodesics both on average and individually.

Dans cet article nous démontrons une formule de trace pour les doubles sommes sur les groupes fuchsiens totalement hyperboliques Γ. Ceci relie les angles d’intersection et les perpendiculaires communes des paires de géodésiques fermées sur Γ/H avec les produits scalaires des carrés de la valeur absolue des fonctions propres du laplacien hyperbolique Δ. Nous arrivons donc à des résultats quantitatifs sur les angles d’intersection et les perpendiculaires communes de ces géodésiques, en moyenne et individuellement.

@article{AIF_1999__49_1_1_0,
     author = {Pitt, Nigel J. E.},
     title = {On pairs of closed geodesics on hyperbolic surfaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {1},
     year = {1999},
     pages = {1-25},
     doi = {10.5802/aif.1667},
     zbl = {0958.11039},
     mrnumber = {2000j:11078},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1999__49_1_1_0}
}
Pitt, Nigel J. E. On pairs of closed geodesics on hyperbolic surfaces. Annales de l'Institut Fourier, Volume 49 (1999) no. 1, pp. 1-25. doi : 10.5802/aif.1667. http://www.numdam.org/item/AIF_1999__49_1_1_0/

[1] A. Beardon, The Geometry of Discrete Groups, Springer, 1983. | MR 698777 | MR 85d:22026 | Zbl 0528.30001

[2] H. Iwaniec, Prime geodesic theorem, J. reine. angew. Math., 349 (1984), 136-159. | MR 743969 | MR 85h:11025 | Zbl 0527.10021

[3] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Biblioteca de la Revista Matemática Iberoamericana (1995). | MR 1325466 | MR 96f:11078 | Zbl 0847.11028

[4] J. Lehner, Discontinuous Groups and Automorphic functions, Amer. Math. Soc. (1964). | MR 164033 | MR 29 #1332 | Zbl 0178.42902

[5] W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on PSL2(ℤ)\ℍ, IHES Publ., 81 (1995 207-237). | Numdam | MR 1361757 | MR 97f:11037 | Zbl 0852.11024

[6] N. Pitt, Talk given at the XIV Escola de Algebra, IMPA, Rio de Janeiro, Aug. 1996.

[7] P. Sarnak, Arithmetic Quantum Chaos, Israel Mathematical Conference Proceedings, 8 (1995). | MR 1321639 | MR 96d:11059 | Zbl 0831.58045

[8] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory, 15 (1982), 229-247. | MR 675187 | Zbl 0499.10021

[9] A. Seger and C. Sogge, Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J., 38 (1989), 669-682. | MR 1017329 | MR 91f:58097 | Zbl 0703.35133

[10] A. Selberg, Collected Papers, Vol. I, Springer, 1989. | MR 92h:01083 | Zbl 0675.10001

[11] S. Zelditch, Selberg Trace Formulae, Pseudodifferential operators and geodesic periods of automorphic forms, Duke Math. J. (1988), 295-344. | Zbl 0646.10024