Trees of the Normalizer of Modular Group in the Picard Group

Year 2018, Volume: 9, 63 - 70, 28.12.2018

Nazlı Yazıcı Gözütok İlgıt Zengin Bahadır Özgür Güler

Abstract

In this study, we investigate trees arising from the imprimitive action of the normalizer of Modular

group in the Picard group on extended rational numbers. We determine the farthest vertex from a given vertex

in hyperbolic paths of minimal lengths. We also include some results of the suborbital graph F_{u,N} related to a

continued fraction representation of a rational number.

Keywords

Normalizer of the modular group, suborbital graphs, continued fraction

References

• Akbaş, M., On Suborbital Graphs for the Modular Group, Bulletin of the London Mathematical Society 33(6)(2001), 647–652.
• Akbaş, M., Bas¸kan, T., Suborbital graphs for the normalizer of 􀀀0(N), Turk J Math, 20(1996), 379–387.
• Beşenk, M., The action of S L(2;C) on hyperbolic 3-space and orbital graphs, Graphs Combin., 34(4)(2018), 545–554.
• Bigg, N.L., White, A.T., Permutation groups and combinatorial structures, London Mathematical Society Lecture Note Series, 33, CUP, Cambridge, 1979.
• Chaichana K, Jaipong P, Suborbital Graphs for Congruence Subgroups of the Extended Modular Group and Continued Fractions, Proceedings of AMM, 20(2015), 86–95.
• Cuyt A. et al., Handbook of Continued Fractions for Special Functions, Springer, New York, 2008.
• Değer AH, Beşenk M, Güler BO, On suborbital graphs and related continued fractions, Appl. Math. Comput., 218(3)(2011), 746–750.
• Değer AH, Vertices of paths of minimal lengths on suborbital graphs, Filomat, 31(4)(2017), 913–923.
• Jones GA, Singerman D, Complex functions: an algebraic and geometric viewpoint, Cambridge University Press, Cambridge, 1987.
• Jones GA, Singerman D, Wicks K, The modular group and generalized Farey graphs. London Math. Soc. Lecture Note Series 160(1991), 316–338.
• Güler, B.Ö . et al., Elliptic elements and circuits in suborbital graphs, Hacet. J. Math. Stat., 40(2)(2011), 203–210.
• Güler, B.Ö ., Kör, T., Şanlı, Z.: Solution to some congruence equations via suborbital graphs. Springerplus, 2016(5)(2016), 1-11.
• Kader, S., Circuits in suborbital graphs for the normalizer. Graphs Combin. 33(6)(2017), 1531–1542.
• Keskin, R., Suborbital graphs for the normalizer 􀀀0(m). European Journal of Combinatorics 27(2)(2006), 193–206.
• Keskin, R., Demirt¨urk, B., On suborbital graphs for the normalizer of 􀀀0(N). The Electronic Journal of Combinatorics 16(1)(2009), 1–18.
• Köroğlu, T., Güler, B.Ö., Şanlı, Z., Suborbital graphs for the Atkin-Lehner group. Turk J Math. 41(2017), 235–243.
• Köroğlu, T., Güler, B.Ö ., Şanlı, Z., Some Generalized Suborbital Graphs. Turk. J. Math. Comput. Sci., 7(2017), 90–95.
• Kushwaha, S.; Sarma, R.; Continued fractions arising from F1;3. Ramanujan J. 46(3)(2018), 605–631.
• Nathanson, M.B., A forest of linear fractional transformations. Int. J. Number Theory 11(4)(2015), 1275–1299.
• Ponton, L., Two trees enumerating the positive rationals. Integers 18A(2018), Paper No. A17, 16 pp.
• Sarma R, Kushwaha S, Krishnan R, Continued fractions arising from F1;2. J. Number Theory 154(2015), 179–200.
• Wall H.S., Analytic Theory of Continued Fractions, first ed., D.Van Nostrand Co, New York, 1948.
• Yazıcı Gözütok, N., Güler, B.Ö ., Suborbital Graphs of the Normalizer of Modular Group in the Picard Group, Iran J Sci Technol Trans Sci 42(4)(2018), 2167–2174.