Relationships between cusp points in the extended modular group and fibonacci numbers

Abstract

Cusp (parabolic) points in the extended modular group (Gamma) over bar are basically the images of infinity under the group elements. This implies that the cusp points of (Gamma) over bar are just rational numbers and the set of cusp points is Q(infinity) = Q boolean OR {infinity} .The Farey graph F is the graph whose set of vertices is Q(infinity) 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 infinity to x in F as < infinity, 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.