Congruence obstructions to pseudomodularity of Fricke groups

TL;DR

This paper investigates conditions under which Fricke groups are pseudomodular, revealing that rational cusp sets must be dense in the finite adeles of nd to be pseudomodular, and showing many such groups are not pseudomodular.

Contribution

It establishes a criterion linking cusp set density in the finite adeles to pseudomodularity of Fricke groups, and demonstrates that infinitely many are not pseudomodular.

Findings

Pseudomodularity requires cusp set density in the finite adeles.

Rational cusp sets are pseudomodular iff dense in finite adeles.

Many Fricke groups with rational cusps are not pseudomodular.

Abstract

A pseudomodular group is a finite coarea nonarithmetic Fuchsian group whose cusp set is exactly $\mathbb{P}^1(\mathbb{Q})$. Long and Reid constructed finitely many of these by considering Fricke groups, i.e., those that uniformize one-cusped tori. We prove that a zonal Fricke group with rational cusps is pseudomodular if and only if its cusp set is dense in the finite adeles of $\mathbb{Q}$. We then deduce that infinitely many such Fricke groups are not pseudomodular.