Exotic projective structures and quasi-Fuchsian space, II
Kentaro Ito
Source: Duke Math. J. Volume 140, Number 1 (2007), 85-109.
Abstract
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
Primary Subjects: 30F40, 57M50
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1190730775
Digital Object Identifier: doi:10.1215/S0012-7094-07-14013-4
Mathematical Reviews number (MathSciNet):
MR2355068
Zentralblatt MATH identifier:
1132.30023
References
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.
Mathematical Reviews (MathSciNet):
MR1411128
Digital Object Identifier: doi:10.1007/s002220050094
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.
Mathematical Reviews (MathSciNet):
MR1815699
Digital Object Identifier: doi:10.2307/2661352
L. Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94--97.
Mathematical Reviews (MathSciNet):
MR0111834
Digital Object Identifier: doi:10.1090/S0002-9904-1960-10413-2
Project Euclid: euclid.bams/1183523461
—, On boundaries of,Teichmüller spaces and on Kleinian groups, I, Ann. of Math. (2) 91 (1970), 570--600.
Mathematical Reviews (MathSciNet):
MR0297992
Digital Object Identifier: doi:10.2307/1970638
F. Bonahon and J.-P. Otal, Variétés hyperboliques à géodésiques arbitrairement courtes, Bull. London Math. Soc. 20 (1988), 255--261.
Mathematical Reviews (MathSciNet):
MR0931188
Digital Object Identifier: doi:10.1112/blms/20.3.255
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.
Mathematical Reviews (MathSciNet):
MR1871491
Project Euclid: euclid.jdg/1090348089
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.
Mathematical Reviews (MathSciNet):
MR2145059
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.
Mathematical Reviews (MathSciNet):
MR0624807
D. M. Gallo, Deforming real projective structures, Ann. Acad. Sci. Fenn. Math. 22 (1997), 3--14.
Mathematical Reviews (MathSciNet):
MR1430390
W. M. Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), 297--326.
Mathematical Reviews (MathSciNet):
MR0882826
Project Euclid: euclid.jdg/1214440978
D. A. Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975), 1--55.
Mathematical Reviews (MathSciNet):
MR0463429
Digital Object Identifier: doi:10.1007/BF02392015
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.
Mathematical Reviews (MathSciNet):
MR2145069
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.
Mathematical Reviews (MathSciNet):
MR0624819
K. Ito, Exotic projective structures and quasi-Fuchsian space, Duke Math. J. 105 (2000), 185--209.
Mathematical Reviews (MathSciNet):
MR1793610
Digital Object Identifier: doi:10.1215/S0012-7094-00-10521-2
Project Euclid: euclid.dmj/1092403841
—, ``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.
Mathematical Reviews (MathSciNet):
MR2258758
—, On continuous extensions of grafting maps, to appear in Trans. Amer. Math. Soc., preprint,\arxivmath/0411133v1[math.GT]
Mathematical Reviews (MathSciNet):
MR2386243
Digital Object Identifier: doi:10.1090/S0002-9947-08-04333-X
T. JøRgensen and A. Marden, Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990), 47--72.
Mathematical Reviews (MathSciNet):
MR1060898
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.
Mathematical Reviews (MathSciNet):
MR1037141
Digital Object Identifier: doi:10.1007/BF01231179
Y. Komori and T. Sugawa, Bers embedding of the Teichmüller space of a once-punctured torus, Conform. Geom. Dyn. 8 (2004), 115--142.
Mathematical Reviews (MathSciNet):
MR2060380
Digital Object Identifier: doi:10.1090/S1088-4173-04-00108-0
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.
Mathematical Reviews (MathSciNet):
MR2229385
Project Euclid: euclid.em/1150476903
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.
Mathematical Reviews (MathSciNet):
MR1873113
A. Marden, The geometry of finitely generated kleinian groups, Ann. of Math. (2) 99 (1974), 383--462.
Mathematical Reviews (MathSciNet):
MR0349992
Digital Object Identifier: doi:10.2307/1971059
K. Matsuzaki and M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford Math. Monogr., Oxford Univ. Press, New York, 1998.
Mathematical Reviews (MathSciNet):
MR1638795
C. T. Mcmullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998), 283--320.
Mathematical Reviews (MathSciNet):
MR1478844
Digital Object Identifier: doi:10.1090/S0894-0347-98-00259-8
S. Nag, The complex analytic theory of Teichmüller spaces, Canadian Math. Soc. Ser. Monogr. Adv. Texts, Wiley, New York, 1988.
Mathematical Reviews (MathSciNet):
MR0927291
D. P. Sullivan, Quasiconformal homeomorphisms and dynamics, II: Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985), 243--260.
Mathematical Reviews (MathSciNet):
MR0806415
Digital Object Identifier: doi:10.1007/BF02392543
Duke Mathematical Journal
