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Relationships between cusp points in the extended modular group and Fibonacci numbers
¨Ozden Koruo˘glu,Sule Kaymak Sarica,Bilal Demir,A. Furkan Kaymak 호남수학회 2019 호남수학학술지 Vol.41 No.3
Cusp (parabolic) points in the extended modular group $\overline{\Gamma}$ are basically the images of infinity under the group elements. This implies that the cusp points of $\overline{\Gamma}$ are just rational numbers and the set of cusp points is $Q_{\infty}=Q\cup\left\{ \infty\right\} .$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) $x=[b_{1},...,b_{n}].$ We get a path from $\infty$ to $x$ in $F$ as \TEXTsymbol{<}$\infty,C_{1},...,C_{n}>$ 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.
¨Ozden Koruo˘glu,Furkan Birol,Recep Sahin,Bilal Demir 호남수학회 2019 호남수학학술지 Vol.41 No.1
We consider the extended generalized Hecke groups $\overline{H}_{3,q}$ generated by $X(z)=-(z-1)^{-1}$, $Y(z)=-(z+\lambda _{q})^{-1}$ with $\lambda _{q}=2\cos (\frac{\pi }{q})$ where $q\geq 3$ an integer.\ In this work, we study the generalized Pell sequences in $\overline{H}_{3,q}.$ Also, we show that the entries of the matrix representation of each element in the extended generalized Hecke Group $\overline{H}_{3,3}$ can be written by using Pell, Pell-Lucas and modified-Pell numbers.
The isomorphism between two fundamental groups by Cayley graphs
˙I. Naci Cang¨ul,A. Sinan C¸ evik,¨Ozden Koruo˘glu 장전수학회 2008 Advanced Studies in Contemporary Mathematics Vol.17 No.1
LetG1 and G2 be two nite groups and letCay(G1,S1) and Cay(G2,S2) be the corresponding Cayley graphs of these groups, respectively. By [2] and [8], one can dene the fundamental groupπ1(Γ,v) by using any connected graph Γ with a fixed vertex v. In this paper we give sufficient conditions for any two fundamental groups which are obtained by Cayley graphs Cay(G1,S1) and Cay(G2,S2) to be isomorphic. At the final part of the paper, we present some examples of this result.