Symbolic Dynamics for the Geodesic Flow on Two-dimensional Hyperbolic Good Orbifolds

TL;DR

This paper develops explicit, geometric cross sections for the geodesic flow on hyperbolic orbifolds, facilitating transfer operator methods for analyzing Maass cusp forms and Selberg zeta functions.

Contribution

It introduces a uniform, explicit, and algorithmic construction of cross sections tailored for transfer operator approaches on hyperbolic orbifolds with nonuniform Fuchsian groups.

Findings

Constructed geometric cross sections for geodesic flow

Enabled transfer operator analysis for Maass cusp forms

Facilitated study of Selberg zeta functions

Abstract

We construct cross sections for the geodesic flow on the orbifolds $\Gamma\backslash H$ which are tailor-made for the requirements of transfer operator approaches to Maass cusp forms and Selberg zeta functions. Here, $H$ denotes the hyperbolic plane and $\Gamma$ is a nonuniform geometrically finite Fuchsian group (not necessarily a lattice, not necessarily arithmetic) which satisfies an additional condition of geometric nature. The construction of the cross sections is uniform, geometric, explicit and algorithmic.