Farey graph and rational fixed points of the extended modular group

Abstract

Fixed points of matrices have many applications in various areas of science and mathematics. The extended modular group Gamma is the group of 2 x 2 matrices with integer entries and determinant +/- 1. There are strong connections between the extended modular group, continued fractions and Farey graph. The Farey graph is a graph with vertex set Q(infinity) = Q boolean OR{8}. In this study we consider the elements in (Gamma) over bar that fix rationals. For a given rational number, we use its Farey neighbours to obtain the matrix representation of the element in (Gamma) over bar that fixes the given rational. Then we express such elements as words in terms of generators using the relations between the Farey graph and continued fractions. Finally we give the new block reduced form of these words which all blocks have Fibonacci numbers entries.