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2\({\pi}\)-Grafting and complex projective structures with generic holonomy

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Abstract

Let S be an oriented closed surface of genus at least two. We show that, given a generic representation \({\rho: \pi_1(S) \to {\rm PSL} (2, \mathbb{C})}\) in the character variety, (\({2\pi}\)-)grafting produces all projective structures on S with holonomy \({\rho}\).

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References

  1. S. Baba. \({2\pi}\)-Grafting and complex projective structures, I. Geometry & Topology, (6)19 (2015), 3233–3287.

  2. Bonahon F.: Geodesic laminations with transverse Hölder distributions. Annales Scientifiques de École Normale Supérieure (4), 30(2), 205–240 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. F. Bonahon. Low-Dimensional Geometry, Student Mathematical Library, Vol. 49. American Mathematical Society, Providence (2009). From Euclidean surfaces to hyperbolic knots, IAS/Park City Mathematical Subseries.

  4. J.F. Brock. The Weil–Petersson metric and volumes of 3-dimensional hyperbolic convex cores. Journal of the American Mathematical Society, (3)16 (2003), 495–535 (electronic).

  5. Boyer S., Zhang X.: On Culler–Shalen seminorms and Dehn filling. Annals of Mathematics (2), 148(3), 737–801 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Calsamiglia G., Deroin B., Francaviglia S.: The oriented graph of multi-graftings in the Fuchsian case. Publicacions Matemàtiques, 58(1), 31–46 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  7. Y.-E. Choi, D. Dumas, and K. Rafi. Grafting rays fellow travel Teichmüller geodesics. International Mathematics Research Notices, Vol. 11 (2012), 2445–2492.

  8. 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). In: London Mathematical Society Lecture Note Series, Vol. 111. Cambridge University Press, Cambridge (1987), pp. 3–92.

  9. Díaz R., Kim I.: Asymptotic behavior of grafting rays. Geometriae Dedicata, 158, 267–281 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Dumas. Complex projective structures, Handbook of Teichmüller theory, Vol. II. In: IRMA Lectures in Mathematics and Theoretical Physics, Vol. 13. European Mathematical Society, Zürich (2009), pp. 455–508.

  11. C.J. Earle. On variation of projective structures. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, NY, 1978), Annals of Mathematics Studies, Vol. 97, Princeton University Press, Princeton, NJ (1981) pp. 87–99.

  12. 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). In: London Mathematical Society Lecture Note Series, Vol. 111. Cambridge University Press, Cambridge (1987), pp. 113–253.

  13. A. Fathi, F. Laudenbach, V. Poénaru, et al. Travaux de Thurston sur les surfaces, Astérisque, Vol. 66. Société Mathématique de France, Paris (1979). Séminaire Orsay, With an English summary.

  14. Gallo D., Kapovich M., Marden A.: The monodromy groups of Schwarzian equations on closed Riemann surfaces. Annals of Mathematics (2), 151(2), 625–704 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. W.M. Goldman. Discontinuous groups and the Euler class. Ph.D. thesis (1980).

  16. Goldman W.M.: Projective structures with Fuchsian holonomy. Journal Differential Geometry, 25(3), 297–326 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  17. Goldman W.M.: Topological components of spaces of representations. Inventiones Mathematicae, 93(3), 557–607 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Gupta. Asymptoticity of grafting and Teichmüller rays. Geometry & Topology, (4)18 (2014), 2127–2188.

  19. Hejhal D.A.: Monodromy groups and linearly polymorphic functions. Acta Mathematica, 135(1), 1–55 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hensel S.W.: Iterated grafting and holonomy lifts of Teichmüller space. Geometriae Dedicata, 155, 31–67 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Heusener and J. Porti. The variety of characters in \({{\rm PSL}_2(\mathbb {C})}\). Boletín de la Sociedad Matemática Mexicana (3), (special issue)10 (2004), 221–237.

  22. Hatcher A., Thurston W.: A presentation for the mapping class group of a closed orientable surface. Topology, 19(3), 221–237 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  23. J.H. Hubbard. The monodromy of projective structures. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, NY, 1978), Annals of Mathematics Studies, Vol. 97. Princeton University Press, Princeton, NJ (1981), pp. 257–275.

  24. K. Ito. Exotic projective structures and quasi-Fuchsian space. II. Duke Mathematical Journal, (1)140 (2007), 85–109.

  25. Kapovich M.: On monodromy of complex projective structures. Inventiones Mathematicae, 119(2), 243–265 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Kapovich. Hyperbolic Manifolds and Discrete Groups, Progress in Mathematics, Vol. 183. Birkhäuser, Boston (2001).

  27. Kulkarni R.S., Pinkall U.: A canonical metric for Möbius structures and its applications. Mathematische Zeitschrift, 216(1), 89–129 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Y. Kamishima and S.P. Tan. Deformation Spaces on Geometric Structures, Aspects of Low-Dimensional Manifolds, Advanced Studies in Pure Mathematics, Vol. 20. Kinokuniya, Tokyo (1992), pp. 263–299.

  29. Y.N. Minsky. The classification of punctured-torus groups. Annals of Mathematics (2), (2)149 (1999), 559–626.

  30. R.C. Penner and J.L. Harer. Combinatorics of Train Tracks, Annals of Mathematics Studies, Vol. 125. Princeton University Press, Princeton (1992).

  31. S. Saul. Notes on the complex of curves. Available at: http://homepages.warwick.ac.uk/~masgar/Maths/notes.pdf.

  32. Tanigawa H.: Grafting, harmonic maps and projective structures on surfaces. Journal of Differential Geometry, 47(3), 399–419 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. W.P Thurston. The geometry and topology of three-manifolds. In: Princeton University Lecture Notes (1978–1981).

  34. W.P. Thurston. Three-Dimensional Geometry and Topology, Vol. 1, Princeton Mathematical Series, Vol. 35. Princeton University Press, Princeto (1997). Edited by Silvio Levy.

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  1. Universität Heidelberg, Heidelberg, Germany

    Shinpei Baba

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  1. Shinpei Baba
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Baba, S. 2\({\pi}\)-Grafting and complex projective structures with generic holonomy. Geom. Funct. Anal. 27, 1017–1069 (2017). https://doi.org/10.1007/s00039-017-0424-9

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