At the end of the previous chapter we noted how important the group PSL(2, ℝ) and its subgroups are to Riemann surface theory. In this chapter we give an introduction to this topic. The plan of the chapter is as follows.

In §§5.1 and 5.2 we discuss basic algebraic properties of PSL(2, ℝ). Much of this is similar to some of the work in Chapter 2 on the group PSL(2, ℂ). In §§5.3 and 5.4 we discuss the hyperbolic metric on the upper half-plane U. With this metric U becomes a model of the hyperbolic plane and PSL(2, ℝ) now acts as a group of isometries. In §§5.6 and 5.7 we introduce Fuchsian groups; these are discrete subgroups of PSL(2, ℝ) and all subgroups of PSL(2, ℝ) which act discontinuously on U come into this class. Thus the groups G of Theorem 4.19.8 are all Fuchsian groups. As these groups are discontinuous groups of hyperbolic isometries they are comparable with lattices which are discontinuous groups of Euclidean isometries. As with lattices, the discontinuity implies the existence of a fundamental region and this is studied in §5.8. As before, we can use Dirichlet regions but now these are hyperbolic polygons and not Euclidean polygons as they were in §2.4. In §5.9, the quotient-spaces of the upper half-plane by Fuchsian groups are proved to be Riemann surfaces.