Exotic Projective Structures and Quasi-fuchsian Space

1. Introduction. Let S be an oriented closed surface of genus g > 1. A projec-tive structure on S is a maximal system of local coordinates modeled on the Riemann sphere C, whose transition functions are Möbius transformations. For a given pro-jective structure on S, we have a pair (f, ρ) of a local homeomorphism f from the universal cover S of S to C, called a developing map, and a group homomorphism ρ of π 1 (S) into PSL 2 (C), called a holonomy representation. Let P (S) denote the space of all (marked) projective structures on S, and let V (S) denote the space of all con-jugacy classes of representations of π 1 (S) into PSL 2 (C). Holonomy representations give a mapping hol : P (S) → V (S), which is called the holonomy mapping. It is known that the map hol is a local homeomorphism (see [13]). The quasi-Fuchsian space QF(S) is the subspace of V (S) consisting of faithful representations whose holonomy images are quasi-Fuchsian groups. In this paper, we investigate the subset Q(S) = hol −1 (QF(S)) of P (S). We say an element of Q(S) is standard if its developing map is injective; otherwise, it is exotic. The set of standard projective structures with fixed underlying complex structure is well known as the image of the Teichmüller space under Bers embedding (see [5]). On the other hand, the existence of exotic projective structures was first shown by Maskit [21]. More investigations of exotic projective structures are found in As we see in Proposition 2.3, each connected component of Q(S) is biholomorphically isomorphic to QF(S). Moreover, as a consequence of the result of Goldman [12], the connected components of Q(S) are in one-to-one correspondence with the set ᏹᏸ Z (S) of integral points of measured laminations. (See Section 2.4 for a precise definition.) We denote by ᏽ λ the component of Q(S) corresponding to λ ∈ ᏹᏸ Z (S), where ᏽ 0 is the component consisting of all standard projective structures. Recently, McMullen [25, Appendix A] discovered the following phenomenon. Theorem 1.1 (McMullen). There exists a sequence of exotic projective structures that converges to a point of the relative boundary ∂ᏽ 0 = ᏽ 0 − ᏽ 0 of ᏽ 0. This phenomenon deeply depends on the following phenomenon in the theory of Kleinian groups: There is a sequence of quasi-Fuchsian groups whose algebraic …