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

Cusp (parabolic) points in the extended modular group □ are basically the images of infinity under the group elements. This implies that the cusp points of □ are just rational numbers and the set of cusp points is Q∞ = Q∪{∞}. The Farey graph F is the graph whose set of vertices is Q∞ 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 ∞ to x in F as < ∞, 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 froms and matrix represantations.