PSL(2,R) are known as Fuchsian groups; they are important in the study of Riemann surfaces. Classification In the following discussion we will always assume that the representing matrix is normalized such that . Non-identity Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and loxodromic, with the hyperbolic ones being a subclass of the loxodromic ones. The classification has both algebraic and geometric significance. Geometrically, the different types result in different transformations of the complex plane, as the figures below illustrate. The four types can be distinguished by looking at the trace . Note that the trace is invariant under conjugation, that is, and so every member of a conjugacy class will have the same trace. Every Möbius transformation can be written such that its representing matrix has determinant one (by multiplying the entries with a suitable scalar). Two Möbius transformations (both not equal to the identity transform) with are conjugate if and only if Parabolic transforms A non-identity Möbius transformation defined by a matrix of determinant one is said to be parabolic if (so the trace is plus or minus 2; either can occur for a given transformation since is determined only up to sign). In fact one of the choices for has the same characteristic polynomial X2−2X+1 as the identity matrix, and is therefore unipotent. A Möbius transform is parabolic if and only if it has exactly one fixed point in the extended complex plane , which happens if and only if it can be defined by a matrix conjugate to which describes a translation in the complex plane. The set of all parabolic Möbius transformations with a given fixed point in , together with the identity, forms a subgroup isomorphic to the group of matrices this is an example of the unipotent radical of a Borel subgroup (of the Möbius group, or of SL(2,C) for the matrix group; the notion is defined for any reductive Lie group). Characteristic constant All non-parabolic transformations have two fixed points and are defined by a matrix conjugate to with the complex number λ not equal to 0, 1 or −1, corresponding to a dilation/rotation through multiplication by the complex number k = λ2, called the characteristic constant or multiplier of the transformation. Elliptic transforms The transformation is said to be elliptic if it can be represented by a matrix whose trace is real with A transform is elliptic if