The determination of parabolic points in modular and extended modular groups by continued fractions

Abstract

The modular group Gamma = PSL(2, Z) is generated by two fractional transformations T(z) = -1/z and S(z) = -1/(z + 1). The extended modular group (Gamma) over bar = PGL(2, Z) is the group obtained by adding the reflection R(z) = 1/(z) over bar to the generators of the modular group F. In this study, we calculate the parabolic points of any given element of the modular and the extended modular groups by simple continued fractions and blocks.