  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.
