on the unit circle. Then F(t) commutes with Γ and so, as for , is the identity on the unit circle. By construction F(t) is an isotopy between the identity and F. This proves injectivity. The proof of surjectivity relies on comparing the hyperbolic metric on D with a word-length metric on Γ. Assuming without loss of generality that 0 lies in the interior of a convex fundamental polygon C and g is an element of Γ, the ray from 0 to g(0)—the hyperbolic geodesic—passes through a succession of translates of C. Each of these is obtained from the previous one by applying a generator of Γ or a fixed product of generators (if successive translates meet in a vertex). It follows that the hyperbolic distance between 0 and g(0) is less than 4g times the word length of g plus twice diameter of the fundamental polygon. Thus the metric on Γ defined by the word length L(g) satisfies for positive constants a and b. Conversely there are positive constants c and d such that Dirichlet polygons Given a point in the upper half-plane H, and a discrete subgroup Γ of that acts freely discontinuously on the upper half-plane, then one can define the Dirichlet polygon as the set of points Here, d is a hyperbolic metric on the upper half-plane. The metric fundamental polygon is more usually called the Dirichlet polygon. This fundamental polygon is a fundamental domain. This fundamental polygon is convex in that the geodesic joining any two points of the polygon is contained entirely inside the polygon. The diameter of F is less than or equal to the diameter of H/Γ. In particular, the closure of F is compact. If Γ has no fixed points in H and H/Γ is compact, then F will have finitely many sides. Each side of the polygon is a geodesic arc. For every side s of the polygon, there is precisely one other side s′ such that for some g in Γ. Thus, this polygon will have an even number of sides. The set of group elements g that join sides to each other are generators of Γ, and there is no smaller set that will generate Γ. The upper half-plane is tiled by the closure of F under the action of Γ. That is, where is the closure of F. Normalised polygon In this section, starting from an arbitrary