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

TL;DR

This paper links Maass cusp forms for Hecke triangle groups to transfer operators and shows the Selberg zeta function as a Fredholm determinant, advancing the understanding of spectral properties of these groups.

Contribution

It introduces a transfer operator framework for Hecke triangle groups and expresses the Selberg zeta function as a Fredholm determinant, unifying spectral and dynamical aspects.

Findings

Maass cusp forms characterized as eigenfunctions of transfer operators

Selberg zeta function expressed as a Fredholm determinant

Connection established between symbolic dynamics and spectral theory

Abstract

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.