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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2007.05941 |
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Table of Contents:
- We study the action of the groups $H(λ)$ generated by the linear fractional transformations $x:z\mapsto -\frac{1}{z} \text{ and }w:z\mapsto z+λ$, where $λ$ 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 $λ=1$ or $λ=2$, and we give an upper bound for the number of orbits for $λ=2$.