Duke Mathematical Journal

Exotic projective structures and quasi-Fuchsian space, II

Kentaro Ito

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Let P(S) be the space of projective structures on a closed surface S of genus g>1, and let Q(S) be the subset of P(S) of projective structures with quasi-Fuchsian holonomy. It is known that Q(S) consists of infinitely many connected components. In this article, we show that the closure of any exotic component of Q(S) is not a topological manifold with boundary and that any two components of Q(S) have intersecting closures

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Duke Math. J. Volume 140, Number 1 (2007), 85-109.

First available in Project Euclid: 25 September 2007

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Zentralblatt MATH identifier

Primary: 30F40: Kleinian groups [See also 20H10] 57M50: Geometric structures on low-dimensional manifolds


Ito, Kentaro. Exotic projective structures and quasi-Fuchsian space, II. Duke Math. J. 140 (2007), no. 1, 85--109. doi:10.1215/S0012-7094-07-14013-4. https://projecteuclid.org/euclid.dmj/1190730775

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  • 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.
  • 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.
  • L. Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94--97.
  • —, On boundaries of,Teichmüller spaces and on Kleinian groups, I, Ann. of Math. (2) 91 (1970), 570--600.
  • F. Bonahon and J.-P. Otal, Variétés hyperboliques à géodésiques arbitrairement courtes, Bull. London Math. Soc. 20 (1988), 255--261.
  • K. Bromberg, personal communication, September 2001.
  • K. Bromberg and J. Holt, Self-bumping of deformation spaces of hyperbolic $3$-manifolds, J. Differential Geom. 57 (2001), 47--65.
  • 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.
  • T. D. Comar, Hyperbolic Dehn surgery and convergence of Kleinian groups (manifolds), Ph.D. dissertation, University of Michigan, Ann Arbor, 1996.
  • 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.
  • D. M. Gallo, Deforming real projective structures, Ann. Acad. Sci. Fenn. Math. 22 (1997), 3--14.
  • W. M. Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), 297--326.
  • D. A. Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975), 1--55.
  • 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.
  • 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.
  • K. Ito, Exotic projective structures and quasi-Fuchsian space, Duke Math. J. 105 (2000), 185--209.
  • —, ``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.
  • —, On continuous extensions of grafting maps, to appear in Trans. Amer. Math. Soc., preprint,\arxivmath/0411133v1[math.GT]
  • T. JøRgensen and A. Marden, Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990), 47--72.
  • 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.
  • Y. Komori and T. Sugawa, Bers embedding of the Teichmüller space of a once-punctured torus, Conform. Geom. Dyn. 8 (2004), 115--142.
  • 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.
  • 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.
  • A. Marden, The geometry of finitely generated kleinian groups, Ann. of Math. (2) 99 (1974), 383--462.
  • K. Matsuzaki and M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford Math. Monogr., Oxford Univ. Press, New York, 1998.
  • C. T. Mcmullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998), 283--320.
  • S. Nag, The complex analytic theory of Teichmüller spaces, Canadian Math. Soc. Ser. Monogr. Adv. Texts, Wiley, New York, 1988.
  • D. P. Sullivan, Quasiconformal homeomorphisms and dynamics, II: Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985), 243--260.