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

TL;DR

This paper investigates how Hecke groups act on certain subsets of quadratic fields, establishing conditions for finite orbits and providing bounds for the number of these orbits.

Contribution

It characterizes when the action of Hecke groups on quadratic field subsets has finitely many orbits, specifically for λ=1 or 2, and provides bounds for the case λ=2.

Findings

Finite number of orbits occurs only for λ=1 or 2.

An upper bound for the number of orbits when λ=2.

The action is infinite for other values of λ.

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 $\lambda=2$.