The transfer operator for the Hecke triangle groups

TL;DR

This paper extends the transfer operator method to analyze the Selberg zeta function for Hecke triangle groups, linking it to continued fractions and period functions, and providing new functional equations.

Contribution

It introduces a transfer operator approach for Hecke triangle groups, connecting geodesic flow, continued fractions, and period functions in a novel way.

Findings

Derived functional equations for transfer operator eigenfunctions.

Linked eigenfunctions at eigenvalue 1 to Lewis-Zagier period functions.

Extended transfer operator techniques to non-arithmetic Hecke groups.

Abstract

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,..., which are non-arithmetic for q \not= 3,4,6. For this we make use of a Poincare map for the geodesic flow on the corresponding Hecke surfaces which has been constructed in arXiv:0801.3951 and which is closely related to the natural extension of the generating map for the so called Hurwitz-Nakada continued fractions. We derive simple 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.