The group PSL(2, R) and its discrete subgroups

In this work, the discussed the properties of the group PSL(2, R) and its discrete subgroups. We considered the relation between this group and hyperbolic geometry. Moreover, we studied algebraic structures of Fuchsian groups which are discrete subgroups of PSL(2, R) and modular group. In the second chapter, some definitions and theorems which will be used later in the work are given. In the fourth chapter, the properties of PSL(2, R) are considered and the action of this group on the upper half plane is studied. In this chapter the upper half plane model of the hyperbolic geometry is constructed. It is seen that the hyperbolic distance and the hyperbolic area are invariant under transformations of PSL(2, R) In the fifth chapter, Fuchsian groups, which are the discrete subgroups of PSL(2, R) are considered. The concept of fundamental regions and tessellations for these groups are given. The quotient spaces of these groups are constructed. The relations between quotient spaces of these groups and compact Riemann surfaces are studied. In the last chapter, the modular group is considered. The generators, fundamental region and representation of the modular group are given.