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Suborbital graphs for the Atkin-Lehner group

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We investigate suborbital graphs for an imprimitive action of the Atkin–Lehner group on a maximal subset of extended rational numbers on which a transitive action is also satisfied. Obtaining edge and some circuit conditions, we examine some combinatorial properties of these graphs. | Turk J Math (2017) 41: 235 – 243 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1602-10 Research Article Suborbital graphs for the Atkin–Lehner group ∗ ¨ ˘ ¨ ur GULER, ¨ Tuncay KORO GLU , Bahadır Ozg¨ Zeynep S ¸ ANLI Department of Mathematics, Karadeniz Technical University, Turkey Received: 02.02.2016 • Accepted/Published Online: 19.04.2016 • Final Version: 03.04.2017 Abstract: We investigate suborbital graphs for an imprimitive action of the Atkin–Lehner group on a maximal subset of extended rational numbers on which a transitive action is also satisfied. Obtaining edge and some circuit conditions, we examine some combinatorial properties of these graphs. Key words: Fuchsian groups, Atkin–Lehner group, group action, suborbital graphs 1. Introduction The idea of a suborbital graph has been used mainly by finite group theorists. In [11], Jones et al. showed that this idea is also useful in the study of the modular group that is a finitely generated Fuchsian group and show that the well-known Farey graph is an example of a suborbital graph. Then similar studies were done for related finitely generated groups.The reader is referred to [2–5,8,9,11– 16] for some relevant previous work on suborbital graphs. Firstly, in [3], it was proved that the elliptic elements in Γ0 (n) correspond to circuits in the subgraph Fu,n of the same order and vice versa. This fact is important because it means that suborbital graphs might have a potential to clarify signature problems taking into account the order of elliptic elements are one of the invariants of signature. Note that it was seen that this relation is just provided unilaterally in [14]. Elliptic elements do not necessarily correspond to circuits of the same order. On the other hand, it is worth noting that these graphs give some number theoretical results about continued fractions and Fibonacci numbers as in [4,8,17]. In the present study, we will continue to .