sphere, the complex plane or the unit disk/upper halfplane. The first important invariant of a compact Riemann surface is its genus, a topological invariant given by half the rank of the Abelian group (which can be identified with the homology group ). The genus is zero if the covering space is the Riemann sphere; one if it is the complex plane; and greater than one if it is the unit disk or upper halfplane. Bihomolomorphisms of the Riemann sphere are just complex Möbius transformations and every non-identity transformation has at least one fixed point, since the corresponding complex matrix always has at least one non-zero eigenvector. Thus if is the Riemann sphere, then X must be simply connected and biholomorphic to the Riemann sphere, the genus zero Riemann surface. When is the complex plane, the group of biholomorphisms is the affine group, the complex Möbius transformations fixing ∞, so the transformations with . The non-identity transformations without fixed points are just those with and , i.e. the non-zero translations. The group Γ can thus be identified with a lattice Λ in C and X with a quotient C/Λ, as described in the section on fundamental polygons in genus one. In the third case when is the unit disk or upper half plane, the group of biholomorphisms consists of the complex Möbius transformations fixing the unit circle or the real axis. In the former case, the transformations correspond to elements of the group in the latter case they correspond to real Möbius transformations, so elements of The study and classification of possible groups Γ that act freely on the unit disk or upper halfplane with compact quotient—the Fuchsian groups of the first kind—can be accomplished by studying their fundamental polygons, as described below. As Poincaré observed, each such polygon has special properties, namely it is convex and has a natural pairing between its sides. These not only allow the group to be recovered but provide an explicit presentation of the group by generators and relations. Conversely Poincaré proved that any such polygon gives rise to a compact Riemann surface; in fact, Poincaré's polygon theorem applied to more general polygons, where the polygon was allowed to have ideal vertices, but his proof was complete only in the compact case, without such vertices. Without assumptions on the convexity of the polygon, complete proofs have been given by Maskit and de Rham, based on