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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.

Reviewer: Gou Nakamura (Toyota)

### MSC:

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

57M50 | General geometric structures on low-dimensional manifolds |

### Keywords:

closed surface### References:

[1] | J. W. Anderson and R. D. Canary, Algebraic limits of Kleinian groups which rearrange the pages of a book , Invent. Math. 126 (1996), 205–214. · Zbl 0874.57012 |

[2] | J. W. Anderson, R. D. Canary, and D. Mccullough, The topology of deformation spaces of Kleinian groups , Ann. of Math. (2) 152 (2000), 693–741. JSTOR: · Zbl 0976.57016 |

[3] | L. Bers, Simultaneous uniformization , Bull. Amer. Math. Soc. 66 (1960), 94–97. · Zbl 0090.05101 |

[4] | -, On boundaries of,Teichmüller spaces and on Kleinian groups, I , Ann. of Math. (2) 91 (1970), 570–600. JSTOR: · Zbl 0197.06001 |

[5] | F. Bonahon and J.-P. Otal, Variétés hyperboliques à géodésiques arbitrairement courtes , Bull. London Math. Soc. 20 (1988), 255–261. · Zbl 0648.53027 |

[6] | K. Bromberg, personal communication, September 2001. |

[7] | K. Bromberg and J. Holt, Self-bumping of deformation spaces of hyperbolic \(3\)-manifolds , J. Differential Geom. 57 (2001), 47–65. · Zbl 1030.57028 |

[8] | R. D. Canary, “Pushing the boundary” in In the Tradition of Ahlfors and Bers, III , Contemp. Math. 355 , Amer. Math. Soc., Providence, 2004, 109–121. · Zbl 1079.57013 |

[9] | T. D. Comar, Hyperbolic Dehn surgery and convergence of Kleinian groups (manifolds) , Ph.D. dissertation, University of Michigan, Ann Arbor, 1996. |

[10] | C. J. Earle, “On variation of projective structures” in Riemann Surfaces and Related Topics (Stony Brook, N.Y., 1978) , Ann. of Math. Stud. 97 , Princeton Univ. Press, Princeton, 1981, 87–99. JSTOR: · Zbl 0474.30036 |

[11] | D. M. Gallo, Deforming real projective structures , Ann. Acad. Sci. Fenn. Math. 22 (1997), 3–14. · Zbl 0868.30043 |

[12] | W. M. Goldman, Projective structures with Fuchsian holonomy , J. Differential Geom. 25 (1987), 297–326. · Zbl 0595.57012 |

[13] | D. A. Hejhal, Monodromy groups and linearly polymorphic functions , Acta Math. 135 (1975), 1–55. · Zbl 0333.34002 |

[14] | J. Holt, “Bumping and self-bumping of deformation spaces” in In the Tradition of Ahlfors and Bers, III , Contemp. Math. 355 , Amer. Math. Soc., Providence, 2004, 269–284. · Zbl 1067.57010 |

[15] | J. H. Hubbard, “The monodromy of projective structures” in Riemann Surfaces and Related Topics (Stony Brook, N.Y., 1978) , Ann. of Math. Stud. 97 , Princeton Univ. Press, Princeton, 1981, 257–275. · Zbl 0475.32008 |

[16] | K. Ito, Exotic projective structures and quasi-Fuchsian space , Duke Math. J. 105 (2000), 185–209. · Zbl 1058.30038 |

[17] | -, “Grafting and components of quasi-Fuchsian projective structures” in Spaces of Kleinian Groups , London Math. Soc. Lecture Note Ser. 329 , Cambridge Univ. Press, Cambridge, 2006. |

[18] | -, On continuous extensions of grafting maps , to appear in Trans. Amer. Math. Soc., preprint,\arxivmath/0411133v1[math.GT] · Zbl 1148.30026 |

[19] | T. JøRgensen and A. Marden, Algebraic and geometric convergence of Kleinian groups , Math. Scand. 66 (1990), 47–72. · Zbl 0738.30032 |

[20] | S. P. Kerckhoff and W. P. Thurston, Noncontinuity of the action of the modular group at Bers’ boundary of Teichmüller space , Invent. Math. 100 (1990), 25–47. · Zbl 0698.32014 |

[21] | Y. Komori and T. Sugawa, Bers embedding of the Teichmüller space of a once-punctured torus , Conform. Geom. Dyn. 8 (2004), 115–142. · Zbl 1051.30039 |

[22] | Y. Komori, T. Sugawa, Y. Yamashita, and M. Wada, Drawing Bers embeddings of the Teichmüller space of once-punctured tori , Experiment. Math. 15 (2006), 51–60. · Zbl 1106.30027 |

[23] | F. Luo, “Some applications of a multiplicative structure on simple loops in surfaces” in Knots, Braids, and Mapping Class Groups –.-Papers Dedicated to Joan S. Birman (New York, 1998) , AMS/IP Stud. Adv. Math. 24 , Amer. Math. Soc., Providence, 2001, 123–129. · Zbl 1011.57007 |

[24] | A. Marden, The geometry of finitely generated kleinian groups , Ann. of Math. (2) 99 (1974), 383–462. JSTOR: · Zbl 0282.30014 |

[25] | K. Matsuzaki and M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups , Oxford Math. Monogr., Oxford Univ. Press, New York, 1998. · Zbl 0892.30035 |

[26] | C. T. Mcmullen, Complex earthquakes and Teichmüller theory , J. Amer. Math. Soc. 11 (1998), 283–320. JSTOR: · Zbl 0890.30031 |

[27] | S. Nag, The complex analytic theory of Teichmüller spaces , Canadian Math. Soc. Ser. Monogr. Adv. Texts, Wiley, New York, 1988. · Zbl 0667.30040 |

[28] | D. P. Sullivan, Quasiconformal homeomorphisms and dynamics, II: Structural stability implies hyperbolicity for Kleinian groups , Acta Math. 155 (1985), 243–260. · Zbl 0606.30044 |

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