A note on the action of Hecke groups on subsets of quadratic fields

Abstract

We study the action of the groups $H(\lambda)$ generated by the linear fractional transformations $x:z\mapsto -\frac{1}{z} \text{ and }w:z\mapsto z+\lambda$, where $\lambda$ is a positive integer, on the subsets $\mathbb Q^*(\sqrt{n})=\{\frac{a+\sqrt n}{c}\;|\;a,b=\frac{a^2-n}{c},c\in\mathbb Z\}$, where $n$ is a square-free integer. We prove that this action has a finite number of orbits if and only if $\lambda=1$ or $\lambda=2$, and we give an upper bound for the number of orbits for $\lambda=2$.