An Extension of Hejhal's Algorithm to Infinite Volume Fundamental Domains

TL;DR

This paper extends Hejhal's algorithm to infinite volume fundamental domains, enabling the numerical computation of Maass forms, eigenvalues, and Hausdorff dimensions for Fuchsian groups of infinite covolume.

Contribution

It introduces a novel approach using flare domains and Fourier expansions to handle infinite volume cases, expanding the applicability of Maass form computations.

Findings

Successfully computed Maass forms for specific infinite volume groups

Demonstrated the algorithm on symmetric Schottky and Hecke groups

Connected eigenvalue computations with Hausdorff dimension estimates

Abstract

This work presents an algorithm for numerically computing Maass forms and their eigenvalues for Fuchsian groups of infinite covolume. By Patterson-Sullivan theory, this has the added benefit of computing Hausdorff dimensions of the limit sets of these groups. To approximate Maass forms, we consider their Fourier expansions in different coordinate systems. To handle infinite volume fundamental domains, we make use of the concept of flare domains. We also develop theory about Fourier expansions in flare domains which mimics the classical theory on expansions with respect to cusps. Finally, we present detailed examples of the algorithm applied to symmetric Schottky groups and infinite volume Hecke groups.