which satisfies similar properties to the original relation. This polygon can in turn be successively modified by reassembling the polygon after cutting it by a diagonal piecewise geodesic segment in its interior. The final polygon has 4g equivalent vertices, with sides that are piecewise geodesic. The sides are labelled by the group elements which give the Möbius transformation to the paired side. In order the labelling is so that Γ is generated by the ai and bi subject to the single relation Using the theory of intersection numbers, it follows that the shape obtained by joining vertices by geodesics is also a proper polygon, not necessarily convex, and is also a fundamental domain with the same group elements giving the pairing. This yields a fundamental polygon with edges given by geodesic segments and with the standard labelling. The abelianisation of Γ, the quotient group , is a free Abelian group with 2g generators. Thus the genus g is a topological invariant. It is easy to see that two Riemann surfaces with the same genus are homeomorphic since as topological space since they are obtained by identifying sides of a 4g-sided polygon—a Euclidean polygon in the Klein model—by diffeomorphisms between paired sides. Applying this construction to the regular 4g-sided polygon allows the Riemann surface to be viewed topologically as a doughnut with g holes, the standard description of oriented surfaces in introductory texts on topology. There are several further results: Two homeomorphic Riemann surfaces are diffeomorphic. Any convex fundamental polygon in genus g has N vertices where 4g ≤ N ≤ 12g – 6. A Dirichlet polygon in genus g has exactly vertices for a dense open set of centres. Every genus g Riemann surface has a Fricke fundamental polygon, i.e. a convex polygon with canonical pairing between sides. (The polygon need not necessarily be a Dirichlet polygon.) After a suitable normalisation and labelling of the generators of the fundamental group, the Fricke polygon is uniquely determined and the real parameters describing it can be used as global real analytic parameters for Teichmüller space in genus g. These results are tied up with the interrelation between homeomorphisms and the fundamental group: this reflects the fact that the mapping class group of a Riemann surface—the group of quasiconformal self-homomorphisms of a Riemann surface H/Γ modulo those homotopic to the identity—can be identified with the outer automorphism group of Γ (the Dehn–Nielsen–Baer theorem).