Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Examples of pleating varieties for twice punctured tori


Authors: Raquel Díaz and Caroline Series
Journal: Trans. Amer. Math. Soc. 356 (2004), 621-658
MSC (2000): Primary 30F40, 20H10, 32G15
Published electronically: September 22, 2003
MathSciNet review: 2022714
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Abstract: We give an explicit description of some pleating varieties (sets with a fixed set of bending lines in the convex hull boundary) in the quasi-Fuchsian space of the twice punctured torus. In accordance with a conjecture of the second author, we show that their closures intersect Fuchsian space in the simplices of minima introduced by Kerckhoff. All computations are done using complex Fenchel-Nielsen coordinates for quasi-Fuchsian space referred to a maximal system of curves.


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  • 1. F. Bonahon and J-P. Otal.
    Laminations mesurées de plissage des variétés hyperboliques de dimension 3,
    preprint, 2001.
  • 2. R. D. Canary, D. B. A. Epstein, and P. Green, Notes on notes of Thurston, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 3–92. MR 903850
  • 3. R. Díaz and C. Series.
    Limits of lines of minima in Thurston's boundary of Teichmüller space, Algebraic and Geometric Topology 3, 207-234, 2003.
  • 4. D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR 903852
  • 5. Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308
  • 6. Frederick Gardiner and Linda Keen, Holomorphic motions and quasi-Fuchsian manifolds, Complex geometry of groups (Olmué, 1998) Contemp. Math., vol. 240, Amer. Math. Soc., Providence, RI, 1999, pp. 159–174. MR 1703557, 10.1090/conm/240/03578
  • 7. Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
  • 8. Robert D. Horowitz, Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math. 25 (1972), 635–649. MR 0314993
  • 9. Linda Keen and Caroline Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993), no. 4, 719–749. MR 1241870, 10.1016/0040-9383(93)90048-Z
  • 10. Linda Keen and Caroline Series, Continuity of convex hull boundaries, Pacific J. Math. 168 (1995), no. 1, 183–206. MR 1331998
  • 11. Linda Keen and Caroline Series, How to bend pairs of punctured tori, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 359–387. MR 1476997, 10.1090/conm/211/02830
  • 12. L. Keen and C. Series.
    Pleating invariants for punctured torus groups,
    Topology, 2003.
  • 13. Linda Keen and Caroline Series, The Riley slice of Schottky space, Proc. London Math. Soc. (3) 69 (1994), no. 1, 72–90. MR 1272421, 10.1112/plms/s3-69.1.72
  • 14. Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, 10.2307/2007076
  • 15. Steven P. Kerckhoff, Lines of minima in Teichmüller space, Duke Math. J. 65 (1992), no. 2, 187–213. MR 1150583, 10.1215/S0012-7094-92-06507-0
  • 16. Y. Komori and C. Series.
    Pleating coordinates for the Earle embedding,
    Ann. de la Fac. des Sciences de Toulouse, Vol. X, 69-105, 2001.
  • 17. Christos Kourouniotis, Complex length coordinates for quasi-Fuchsian groups, Mathematika 41 (1994), no. 1, 173–188. MR 1288062, 10.1112/S0025579300007270
  • 18. Irwin Kra, On lifting Kleinian groups to 𝑆𝐿(2,𝐶), Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 181–193. MR 780044
  • 19. Caroline Series, Lectures on pleating coordinates for once punctured tori, Sūrikaisekikenkyūsho Kōkyūroku 1104 (1999), 30–90. Hyperbolic spaces and related topics (Japanese) (Kyoto, 1998). MR 1744472
  • 20. Caroline Series, On Kerckhoff minima and pleating loci for quasi-Fuchsian groups, Geom. Dedicata 88 (2001), no. 1-3, 211–237. MR 1877217, 10.1023/A:1013171204254
  • 21. C. Series, Limits of quasifuchsian groups with small bending, preprint 2002. arXiv:mathGT/0209190
  • 22. Ser Peow Tan, Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures, Internat. J. Math. 5 (1994), no. 2, 239–251. MR 1266284, 10.1142/S0129167X94000140
  • 23. William P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 91–112. MR 903860
  • 24. William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975

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Additional Information

Raquel Díaz
Affiliation: Departamento de Geometría y Topología, Facultad Matemáticas, Universidad Complutense, 28040 Madrid, Spain
Email: radiaz@mat.ucm.es

Caroline Series
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: cms@maths.warwick.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03179-9
Received by editor(s): August 21, 2001
Received by editor(s) in revised form: July 11, 2002
Published electronically: September 22, 2003
Article copyright: © Copyright 2003 American Mathematical Society