Notes on projective structures and Kleinian groups

Abstract

Throughout this paper, C will denote the complex plane, C—C\J {oo} the number sphere, and D= {z: \z PSL(2, C) and a locally univalent holomorphic Mequivariant map (the developing map)f: D-*C> so that foy=M(y)of for all γ G Γ (see [2, ch.9)]). Let M(Y) denote the image of Γ by M. The kernel of a projective structure means ker(M), and a projective structure is called faithful if its kernel is trivial. Two projective structures (M^/i) and (M2,/2) are said to be equivalent if and only if there is a £GPSL(2, C) SO that fi=g°f2 (and hence M1=^gM2g~ ). It is well known that the space of equivalence classes of projective structures is in one-to-one correspondence with the affine space of holomorphic quadratic differentials on i?, Q(i?). We choose an origin in the space of quadratic differentials by fixing an equivalence class of projective structures. In this case the Fuchsian equivalence class will be denoted by OeQ(i?), making Q(R) a vector space. Now the mapping from equivalence classes of projective structures to quadratic differentials is readily expressed: P=(M,f) corresponds to the quadratic differential Sf^ Q(R), where Sf is the Schwarzian derivative of /: J9-»C. This indeed determines a map from the equivalence classes of projective structures to Q(R)) for any ^ePSL(2, C), Sgof=Sfy implying that equivalent projective structures correspond to the same quadratic differential. This map is actually a bijection (see either [2, ch.9] or [4, §11.3]), and we will identify equivalent projective structures and use the identification with Q(R) implicitly. Thus PQ means a representative of the equivalence class of projectivje structures corresponding to Q G