## 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

closed surface

### Citations:

Zbl 0595.57012; Zbl 0890.30031; Zbl 1058.30038
Full Text:

### References:

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