The Determination of Parabolic Points in Modular and Extended Modular Groups by Continued Fractions

The modular group Γ = PSL(2, Z) is generated by two fractional transformations T(z) = -1/z and S(z) = -1/(z + 1): The extended modular group bar{Γ} = PGL(2, Z) is the group obtained by adding the reection R(z) = 1/ bar{z} to the generators of the modular group 􀀀: 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.