Congruence obstructions to pseudomodularity of Fricke groups

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.