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

2022/05/24 by Alexander Karlovitz, Karlovitz, Alexander · 1 citation
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2205.12107

openalex publication_date 2022/05/24 · openalex created_date 2022/05/27 · openalex updated_date 2026/07/28

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.

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