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

Year 2022, Volume: 3 Issue: 2, 160 - 171, 28.07.2022

Bilal Demir

https://doi.org/10.54974/fcmathsci.1082199

Cited By: 1

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.

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