A Note on Transitive Action of the Extended Modular Group on Rational Numbers

Abstract

The extended modular group $\overline{\Gamma}$ is isomorphic to the amalgamated free product of two dihedral groups $D_2$ and $D_3$ with amalgamation $\mathbb{Z}_2$. This group acts on rational numbers transitively. In this study, we obtain elements in the extended modular group that are mappings between given two rationals. Also, we express these elements as a word in generators. We use interesting relations between continued fractions and the Farey graph.

Keywords

• Extended modular group
• continued fractions
• Farey graph.

References

1. Beardon A.F., Algebra and Geometry, Cambridge University Press, 2005.
2. Beardon A.F., Hockman M., Short I., Geodesic continued fractions, The Michigan Mathematical Journal, 61(1), 133-150, 2012.
3. Fine B., Trace classes and quadratic forms in the modular group, Canadian Mathematical Bulletin, 37(2), 202-212, 1994.
4. Gözütok N.Y., Güler B.Ö., Elliptic elements of a subgroup of the normalizer and circuits in orbital graphs, Applications and Applied Mathematics: An International Journal, 14(3), 11-21, 2019.
5. Güler B.Ö., Beşenk M., Değer A., Kader S., Elliptic elements and circuits in suborbital graphs, Hacettepe Journal of Mathematics and Statistics, 40(2), 203-210, 2011.
6. Jones G.A., Singerman D., Complex Functions: An Algebraic and Geometric Viewpoint, Cambridge University Press, 1987.
7. Koruoğlu Ö, Sarıca Ş., Demir B., Kaymak A.F., Relationships between cusp points in the extended modular group and Fibonacci numbers, Honam Mathematical Journal, 41(3), 569-579, 2019.
8. Koruoğlu Ö., Şahin R., İkikardeş S., Trace classes and fixed points for the extended modular group, Turkish Journal of Mathematics, 32(1), 11-19, 2008.

9. Koruoğlu Ö., Şahin R., Generalized Fibonacci sequences related to the extended Hecke groups and an application to the extended modular group, Turkish Journal of Mathematics, 34(3), 325-332, 2010.
10. Morier-Genoud S., Ovsienko V., Farey boat: Continued fractions and triangulations, modular group and polygon dissections, Jahresbericht der Deutschen Mathematiker-Vereinigung, 121(2), 91-136, 2019.
11. Rosen D., A class of continued fractions associated with certain properly discontinuous groups, Duke Mathematical Journal, 21(3), 549-563, 1954.
12. Short I., Walker M., Even-integer continued fractions and the Farey tree, Springer Proceedings in Mathematics and Statistics, 159, 287-300, 2016.
13. Short I., Walker M., Geodesic Rosen continued fractions, The Quarterly Journal of Mathematics, 67(4), 519-549, 2016.
14. [14] Wicks K., Jones G.A., Singerman D., The modular group and generalized Farey graphs, London Mathematical Society Lecture Note Series, 160, 316-338, 1991.