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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)
  • Received : 2018.12.13
  • Accepted : 2019.03.06
  • Published : 2019.09.25

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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