hyperbolic plane). As such, Möbius transformations play an important role in the theory of Riemann surfaces. The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). A particularly important discrete subgroup of the Möbius group is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations. Möbius transformations can be more generally defined in spaces of dimension n > 2 as the bijective conformal orientation-preserving maps from the n-sphere to the n-sphere. Such a transformation is the most general form of conformal mapping of a domain. According to Liouville's theorem a Möbius transformation can be expressed as a composition of translations, similarities, orthogonal transformations and inversions. Definition The general form of a Möbius transformation is given by where a, b, c, d are any complex numbers satisfying . If , the rational function defined above is a constant since and is thus not considered a Möbius transformation. In case , this definition is extended to the whole Riemann sphere by defining If , we define Thus a Möbius transformation is always a bijective holomorphic function from the Riemann sphere to the Riemann sphere. The set of all Möbius transformations forms a group under composition. This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps. The Möbius group is then a complex Lie group. The Möbius group is usually denoted as it is the automorphism group of the Riemann sphere. Fixed points Every non-identity Möbius transformation has two fixed points on the Riemann sphere. Note that the fixed points are counted here with multiplicity; the parabolic transformations are those where the fixed points coincide. Either or both of these fixed points may be the point at infinity. Determining the fixed points The fixed points of the transformation are obtained by solving the fixed point equation f(γ) = γ. For c ≠ 0, this has two roots obtained by expanding this equation to and applying the quadratic formula. The roots are with discriminant Parabolic transforms have coincidental fixed points due to zero discriminant. For c nonzero and nonzero discriminant the transform is elliptic or hyperbolic. When c = 0, the quadratic equation degenerates into a linear equation and the transform is linear. This corresponds to the situation that one of