Comptes Rendus
Group theory/Number theory
Coset diagrams of the modular group and continued fractions
Comptes Rendus. Mathématique, Volume 357 (2019) no. 8, pp. 655-663.

The coset diagram for each orbit under the action of the modular group on Q(n)=Q(n){} contains a circuit Ci. For any αQ(n), the path leading to the circuit Ci and the circuit itself are obtained through continued fractions in this paper. We show that the structure of the continued fractions of a reduced quadratic irrational element is weaved with the structure or type of the circuit. The three types of circuits of the action of V4 on Q(n) are also interconnected with the structure of continued fractions. The action of the modular group on Q(5) is chosen specifically because a circuit of it is related to the ratio of the Fibonacci numbers being the solution to the continued fractions of the golden ratio.

Le diagramme des classes de chaque orbite de l'action du groupe modulaire sur Q(n)=Q(n){} contient un circuit Ci. Dans cette Note, pour tout αQ(n), le chemin menant au circuit Ci et le circuit lui-même sont décrits en termes de fractions continues. Nous montrons que la structure des fractions continues des nombres quadratiques irrationnels réduits est liée à la structure ou au type du circuit. Les trois types de circuits de l'action de V4 sur Q(n) sont également reliés à la structure des fractions continues. L'action du groupe modulaire sur Q(5) est choisie précisément, car un de ses circuits est lié au fait que les rapports des nombres de Fibonacci sont les convergents de la fraction continue du nombre d'or.

Published online:
DOI: 10.1016/j.crma.2019.07.002

Ayesha Rafiq 1; Qaiser Mushtaq 2

1 Institute of Space Technology, Islamabad, Pakistan
2 The Islamia University of Bahawalpur, Bahawalpur, Pakistan
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     title = {Coset diagrams of the modular group and continued fractions},
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     pages = {655--663},
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     year = {2019},
     doi = {10.1016/j.crma.2019.07.002},
     language = {en},
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Ayesha Rafiq; Qaiser Mushtaq. Coset diagrams of the modular group and continued fractions. Comptes Rendus. Mathématique, Volume 357 (2019) no. 8, pp. 655-663. doi : 10.1016/j.crma.2019.07.002.

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Cited by Sources:

This work was presented at the 13th International Pure Mathematics Conference 2012, Islamabad, Pakistan.

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