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Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

Published online by Cambridge University Press:  30 November 2011

Goethe-Universität Frankfurt, Institut für Mathematik, 60325 Frankfurt (Main), Germany (email:
ETH Zürich, Departement Mathematik, Rämistrasse 101, 8092 Zürich, Switzerland (email:


We characterize Maass cusp forms for any cofinite Hecke triangle group as 1-eigenfunctions of appropriate regularity of a transfer operator family. This transfer operator family is associated to a certain symbolic dynamics for the geodesic flow on the orbifold arising as the orbit space of the action of the Hecke triangle group on the hyperbolic plane. Moreover, we show that the Selberg zeta function is the Fredholm determinant of the transfer operator family associated to an acceleration of this symbolic dynamics.

Research Article
Copyright © Cambridge University Press 2011

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[BLZ]Bruggeman, R., Lewis, J. and Zagier, D.. Period functions for Maass wave forms. II: cohomology. Preprint.Google Scholar
[BM09]Bruggeman, R. W. and Mühlenbruch, T.. Eigenfunctions of transfer operators and cohomology. J. Number Theory 129(1) (2009), 158181.CrossRefGoogle Scholar
[Bru97]Bruggeman, R.. Automorphic forms, hyperfunction cohomology, and period functions. J. Reine Angew. Math. 492 (1997), 139.Google Scholar
[CM99]Chang, C.-H. and Mayer, D.. The transfer operator approach to Selberg’s zeta function and modular and Maass wave forms for PSL(2,Z). Emerging Applications of Number Theory (Minneapolis, MN, 1996) (The IMA Volumes in Mathematics and its Applications, 109). Springer, New York, 1999, pp. 73141.CrossRefGoogle Scholar
[CM01a]Chang, C.-H. and Mayer, D.. Eigenfunctions of the transfer operators and the period functions for modular groups. Dynamical, Spectral, and Arithmetic Zeta Functions (San Antonio, TX, 1999) (Contemporary Mathematics, 290). American Mathematical Society, Providence, RI, 2001, pp. 140.Google Scholar
[CM01b]Chang, C.-H. and Mayer, D.. An extension of the thermodynamic formalism approach to Selberg’s zeta function for general modular groups. Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, Berlin, 2001, pp. 523562.CrossRefGoogle Scholar
[DH07]Deitmar, A. and Hilgert, J.. A Lewis correspondence for submodular groups. Forum Math. 19(6) (2007), 10751099.CrossRefGoogle Scholar
[DIPS85]Deshouillers, J.-M., Iwaniec, H., Phillips, R. S. and Sarnak, P.. Maass cusp forms. Proc. Natl. Acad. Sci. USA 82(11) (1985), 35333534.CrossRefGoogle ScholarPubMed
[Efrat]Efrat, I.. Dynamics of the continued fraction map and the spectral theory of SL(2,Z). Invent. Math. 114(1) (1993), 207218; MR 1235024(94h:11052).CrossRefGoogle Scholar
[Fis87]Fischer, J.. An Approach to the Selberg Trace Formula via the Selberg Zeta-function (Lecture Notes in Mathematics, 1253). Springer, Berlin, 1987.CrossRefGoogle Scholar
[FMM07]Fraczek, M., Mayer, D. and Mühlenbruch, T.. A realization of the Hecke algebra on the space of period functions for Γ0(n). J. Reine Angew. Math. 603 (2007), 133163.Google Scholar
[Fri96]Fried, D.. Symbolic dynamics for triangle groups. Invent. Math. 125(3) (1996), 487521.CrossRefGoogle Scholar
[Gro55]Grothendieck, A.. Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 1955(16) (1955), 140.Google Scholar
[Gro56]Grothendieck, A.. La théorie de Fredholm. Bull. Soc. Math. France 84 (1956), 319384.CrossRefGoogle Scholar
[Hej83]Hejhal, D. A.. The Selberg Trace Formula for PSL(2,R). Vol. 2 (Lecture Notes in Mathematics, 1001). Springer, Berlin, 1983.CrossRefGoogle Scholar
[Hej92]Hejhal, D. A.. Eigenvalues of the Laplacian for Hecke triangle groups. Mem. Amer. Math. Soc. 97(469) (1992), vi+165.Google Scholar
[HMM05]Hilgert, J., Mayer, D. and Movasati, H.. Transfer operators for Γ0(n) and the Hecke operators for the period functions of . Math. Proc. Cambridge Philos. Soc. 139(1) (2005), 81116.CrossRefGoogle Scholar
[Lew97]Lewis, J.. Spaces of holomorphic functions equivalent to the even Maass cusp forms. Invent. Math. 127 (1997), 271306.CrossRefGoogle Scholar
[LZ01]Lewis, J. and Zagier, D.. Period functions for Maass wave forms. I. Ann. of Math. (2) 153(1) (2001), 191258.CrossRefGoogle Scholar
[May76]Mayer, D.. On a ζ function related to the continued fraction transformation. Bull. Soc. Math. France 104(2) (1976), 195203.CrossRefGoogle Scholar
[May90]Mayer, D.. On the thermodynamic formalism for the Gauss map. Comm. Math. Phys. 130(2) (1990), 311333.CrossRefGoogle Scholar
[May91]Mayer, D.. The thermodynamic formalism approach to Selberg’s zeta function for PSL(2,Z). Bull. Amer. Math. Soc. (N.S.) 25(1) (1991), 5560.CrossRefGoogle Scholar
[MMS10]Mayer, D., Mühlenbruch, T. and Strömberg, F.. The transfer operator for the Hecke triangle groups. arXiv:0912.2236, 2010.Google Scholar
[MS08]Mayer, D. and Strömberg, F.. Symbolic dynamics for the geodesic flow on Hecke surfaces. J. Mod. Dyn. 2(4) (2008), 581627.Google Scholar
[Mor97]Morita, T.. Markov systems and transfer operators associated with cofinite Fuchsian groups. Ergod. Th. & Dynam. Sys. 17(5) (1997), 11471181.CrossRefGoogle Scholar
[PS85b]Phillips, R. S. and Sarnak, P.. The Weyl theorem and the deformation of discrete groups. Comm. Pure Appl. Math. 38(6) (1985), 853866.CrossRefGoogle Scholar
[PS85a]Phillips, R. S. and Sarnak, P.. On cusp forms for co-finite subgroups of PSL(2,R). Invent. Math. 80(2) (1985), 339364.CrossRefGoogle Scholar
[Poh10]Pohl, A.. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. arXiv:1008.0367v1, 2010.Google Scholar
[Pol91]Pollicott, M.. Some applications of thermodynamic formalism to manifolds with constant negative curvature. Adv. Math. 85 (1991), 161192.CrossRefGoogle Scholar
[Rue76]Ruelle, D.. Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34(3) (1976), 231242.CrossRefGoogle Scholar
[Rue94]Ruelle, D.. Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (CRM Monograph Series, 4). American Mathematical Society, Providence, RI, 1994.Google Scholar
[Sel56]Selberg, A.. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.) 20 (1956), 4787.Google Scholar
[Ser85]Series, C.. The modular surface and continued fractions. J. Lond. Math. Soc. (2) 31(1) (1985), 6980.CrossRefGoogle Scholar