The transfer operator for the Hecke triangle groups
Dieter Mayer Tobias Mühlenbruch Fredrik Strömberg
In this paper we extend the transfer operator approach to Selberg's zeta function for cofinite Fuchsian groups to the Hecke triangle groups $G_q,\, q=3,4,\ldots$, which are non-arithmetic for $q\not= 3,4,6$. For this we make use of a Poincar\'e map for the geodesic flow on the corresponding Hecke surfaces, which has been constructed in [13], and which is closely related to the natural extension of the generating map for the so-called Hurwitz-Nakada continued fractions. We also derive functional equations for the eigenfunctions of the transfer operator which for eigenvalues $\rho =1$ are expected to be closely related to the period functions of Lewis and Zagier for these Hecke triangle groups.
keywords: $\lambda_q$-continued fractions Ruelle and Selberg zeta function. Hecke triangle groups transfer operator

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