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Exotic projective structures and quasi-Fuchsian space. II. (English) Zbl 1132.30023

The author considers in this paper how the space of projective structures on a closed surface includes its subset of projective structures with quasi-Fuchsian holonomy. Let \(S\) be an oriented closed surface of genus \(g>1\). Let \(P(S)\) be the space of marked projective structures on \(S\) and \(R(S)\) the set of conjugacy classes of representations \(\rho:\pi_1(S)\to \text{PSL}_2(\mathbb{C})\). Then we have the holonomy mapping hol: \(P(S)\to R(S)\) and \(Q(S)=\text{hol}^{-1}(\mathcal{QF})\), where \(\mathcal{QF}\) denotes the quasi-Fuchsian space. Let \(\mathcal{ML}_{\mathbb{N}}\) be the set of integral points of measured laminations. By the result of W. M. Goldman, there exists a correspondence between connected components of \(Q(S)\) and integral points of measured laminations [J. Differ. Geom. 25, 297-326 (1987; Zbl 0595.57012)]. An element of \(Q(S)\) is said to be standard if its developing map is injective, otherwise exotic. C. T. McMullen showed that there exists a sequence of exotic projective structures which converges to a point on the boundary of the standard component \(\mathcal{Q}_0\) [J. Am. Math. Soc. 11, No. 2, 283–320 (1998; Zbl 0890.30031)]. Denote by \(\{Y_n\}\) a bi-infinite convergent sequence of projective structures such that \(Y_n\to Y_{\infty}\in \partial \mathcal{Q}_0\) \((| n| \to \infty)\) for a nonzero \(\lambda\in \mathcal{ML}_{\mathbb{N}}\), where \(Y_n\) are exotic for all large \(| n| \). It was shown by the author that \(Y_n\in \mathcal{Q}_{\lambda}\) for all large \(| n| \), and therefore \(\overline{\mathcal{Q}_0}\cap \overline{\mathcal{Q}_{\lambda}}\neq \emptyset\) [Duke Math. J. 105, No. 2, 185–209 (2000; Zbl 1058.30038)]. Let \(Z_{\infty}=\text{Gr}_{\mu}(Y_{\infty})\) be the grafting of \(Y_{\infty}\) along \(\mu\in \mathcal{ML}_{{\mathbb N}}\), where \(\mu\) is nonzero and has no curve in the support which is isotopic to one in the support of \(\lambda\). Owing to the local homeomorphicity of the map hol, there exists a bi-infinite convergent sequence \(Z_n\to Z_{\infty}\in \partial \mathcal{Q}_{\mu}\) \((| n| \to \infty)\) such that \(\text{hol}(Z_n)=\text{hol}(Y_n)\) for all large \( | n| \). In this paper the author proves that \(Z_n\in \mathcal{Q}_{(\lambda,\mu)_{\sharp}}\) and \(Z_{-n}\in {\mathcal Q}_{(\lambda,\mu)_{\flat}}\) for all large \(n\), where \((\lambda,\mu)_{\sharp}\) and \((\lambda,\mu)_{\flat}\) are elements of \(\mathcal{ML}_{{\mathbb N}}\) determined by the two ways of resolving intersecting points of the realizations of \(\lambda\) and \(\mu\). It is proved by constructing regular neighborhoods of realizations of \((\lambda,\mu)_{\sharp}\) and \((\lambda,\mu)_{\flat}\) in \(Z_n\) which contain exactly two components of the pullbacks \(\Lambda_{Z_n}\). This result gives several interesting properties of \(\mathcal{Q}_{\lambda}\) as follows: \(\mathcal{Q}_{\lambda}\) self-bumps for any nonzero \(\lambda \in \mathcal{ML}_{{\mathbb N}}\); \(\overline{\mathcal{Q}_{\lambda}}\cap \overline{\mathcal{Q}_{\mu}}\neq \emptyset\) for any \(\lambda, \mu \in \mathcal{ML}_{{\mathbb N}}\), and so on.

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57M50 General geometric structures on low-dimensional manifolds

Keywords:

closed surface
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References:

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