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http://dx.doi.org/10.5831/HMJ.2019.41.3.569

RELATIONSHIPS BETWEEN CUSP POINTS IN THE EXTENDED MODULAR GROUP AND FIBONACCI NUMBERS  

Koruoglu, Ozden (Department of Mathematics, Balikesir University, Necatibey Faculty of Education)
Sarica, Sule Kaymak (Department of Mathematics, Balikesir University, Institue of Science)
Demir, Bilal (Department of Mathematics, Balikesir University, Necatibey Faculty of Education)
Kaymak, A. Furkan (Department of Computer Engineering, Ege University, Engineering Faculty)
Publication Information
Honam Mathematical Journal / v.41, no.3, 2019 , pp. 569-579 More about this Journal
Abstract
Cusp (parabolic) points in the extended modular group ${\bar{\Gamma}}$ are basically the images of infinity under the group elements. This implies that the cusp points of ${\bar{\Gamma}}$ are just rational numbers and the set of cusp points is $Q_{\infty}=Q{\cup}\{{\infty}\}$.The Farey graph F is the graph whose set of vertices is $Q_{\infty}$ and whose edges join each pair of Farey neighbours. Each rational number x has an integer continued fraction expansion (ICF) $x=[b_1,{\cdots},b_n]$. We get a path from ${\infty}$ to x in F as $<{\infty},C_1,{\cdots},C_n>$ for each ICF. In this study, we investigate relationships between Fibonacci numbers, Farey graph, extended modular group and ICF. Also, we give a computer program that computes the geodesics, block forms and matrix represantations.
Keywords
extended modular group; modular group; Farey graph; Fibonacci numbers;
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