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Skinning maps are finite-to-one

Abstract

We show that Thurston’s skinning maps of Teichmüller space have finite fibers. The proof centers around a study of two subvarieties of the \({{\rm SL}_2(\mathbb{C})}\) character variety of a surface—one associated with complex projective structures, and the other associated with a 3-manifold. Using the Morgan–Shalen compactification of the character variety and author’s results on holonomy limits of complex projective structures, we show that these subvarieties have only a discrete set of intersections.

Along the way, we introduce a natural stratified Kähler metric on the space of holomorphic quadratic differentials on a Riemann surface and show that it is symplectomorphic to the space of measured foliations. Mirzakhani has used this symplectomorphism to show that the Hubbard–Masur function is constant; we include a proof of this result. We also generalize Floyd’s theorem on the space of boundary curves of incompressible, boundary-incompressible surfaces to a statement about extending group actions on \({\Lambda}\) -trees.

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References

  1. Alexander H.: Continuing 1-dimensional analytic sets. Math. Ann., 191, 143–144 (1971)

    MATH  MathSciNet  Article  Google Scholar 

  2. Alperin, R. & Bass, H., Length functions of group actions on \({\Lambda}\)-trees, in Combinatorial Group Theory and Topology (Alta, UT, 1984), Ann. of Math. Stud., 111, pp. 265–378. Princeton Univ. Press, Princeton, NJ, 1987.

  3. Athreya J., Bufetov A., Eskin A., Mirzakhani M.: Lattice point asymptotics and volume growth on Teichmüller space. Duke Math. J., 161, 1055–1111 (2012)

    MATH  MathSciNet  Article  Google Scholar 

  4. Bers L.: On boundaries of Teichmüller spaces and on Kleinian groups. I. Ann. of Math., 91, 570–600 (1970)

    MATH  MathSciNet  Article  Google Scholar 

  5. Bers, L., Spaces of Kleinian groups, in Several Complex Variables, I (College Park, MD, 1970), pp. 9–34. Springer, Berlin–Heidelberg, 1970.

  6. Bonahon F.: Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form. Ann. Fac. Sci. Toulouse Math., 5, 233–297 (1996)

    MATH  MathSciNet  Article  Google Scholar 

  7. Bowditch B. H.: Group actions on trees and dendrons.. Topology, 37, 1275–1298 (1998)

    MATH  MathSciNet  Article  Google Scholar 

  8. Bridson, M. R. & Haefliger, A., Metric Spaces of Non-Positive Curvature. Grundlehren der Mathematischen Wissenschaften, 319. Springer, Berlin–Heidelberg, 1999.

  9. Brown K. S.: Trees, valuations, and the Bieri–Neumann–Strebel invariant. Invent. Math., 90, 479–504 (1987)

    MATH  MathSciNet  Article  Google Scholar 

  10. Chirka, E. M., Regularity of the boundaries of analytic sets. Mat. Sb., 117(159) (1982), 291–336, 431 (Russian); English translation in Math. USSR–Sb., 45 (1983), 291–335.

  11. Chirka, E. M., Complex Analytic Sets. Mathematics and its Applications (Soviet Series), 46. Kluwer, Dordrecht, 1989.

  12. Chiswell, I., Introduction to \({\Lambda}\) -Trees. World Scientific, River Edge, NJ, 2001.

  13. Culler M.: Lifting representations to covering groups. Adv. Math., 59, 64–70 (1986)

    MATH  MathSciNet  Article  Google Scholar 

  14. Culler M., Morgan J. W.: Group actions on R-trees. Proc. London Math. Soc., 55, 571–604 (1987)

    MATH  MathSciNet  Article  Google Scholar 

  15. Culler M., Shalen P. B.: Varieties of group representations and splittings of 3-manifolds. Ann. of Math., 117, 109–146 (1983)

    MATH  MathSciNet  Article  Google Scholar 

  16. Douady A., Hubbard J.: On the density of Strebel differentials. Invent. Math., 30, 175–179 (1975)

    MATH  MathSciNet  Article  Google Scholar 

  17. Dumas, D., Holonomy limits of complex projective structures. Preprint, 2011. arXiv:1105.5102 [math.DG].

  18. Dumas D., Kent R. P.: IV, Bers slices are Zariski dense. J. Topol., 2, 373–379 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  19. Dumas D., Kent R. P.: Slicing, skinning, and grafting. Amer. J. Math., 131, 1419–1429 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  20. Dumas, D. & Kent, R. P., Experiments with skinning maps. In preparation.

  21. Fathi, A., Laudenbach, F. & Poénaru, V. (Eds.), Travaux de Thurston sur les surfaces. Astérisque, 66. Soc. Math. de France, Paris, 1979.

  22. Floyd, W. J., Incompressible surfaces in 3-manifolds: the space of boundary curves, in Low-Dimensional Topology and Kleinian Groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., 112, pp. 131–143. Cambridge Univ. Press, Cambridge, 1986.

  23. Floyd W. J., Oertel U.: Incompressible surfaces via branched surfaces. Topology, 23, 117–125 (1984)

    MATH  MathSciNet  Article  Google Scholar 

  24. Gallo D., Kapovich M., Marden A.: The monodromy groups of Schwarzian equations on closed Riemann surfaces. Ann. of Math., 151, 625–704 (2000)

    MATH  MathSciNet  Article  Google Scholar 

  25. Gardiner F. P.: Measured foliations and the minimal norm property for quadratic differentials. Acta Math., 152, 57–76 (1984)

    MATH  MathSciNet  Article  Google Scholar 

  26. Gaster, J., A family of non-injective skinning maps with critical points. To appear in Trans. Amer. Math. Soc.

  27. Goresky, M. & MacPherson, R., Stratified Morse Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 14. Springer, Berlin–Heidelberg, 1988.

  28. Gravett K. A. H.: Ordered abelian groups. Q. J. Math., 7, 57–63 (1956)

    MATH  MathSciNet  Article  Google Scholar 

  29. Hardt R. M.: Stratification of real analytic mappings and images. Invent. Math., 28, 193–208 (1975)

    MATH  MathSciNet  Article  Google Scholar 

  30. Hatcher A. E.: On the boundary curves of incompressible surfaces. Pacific J. Math., 99, 373–377 (1982)

    MATH  MathSciNet  Article  Google Scholar 

  31. Heusener M., Porti J.: The variety of characters in \({{\rm PSL_2} (\mathbb{C})}\). Bol. Soc. Mat. Mex., 10, 221–237 (2004)

    MATH  MathSciNet  Google Scholar 

  32. Hironaka, H., Subanalytic sets, in Number Theory, Algebraic Geometry and Commutative Algebra, pp. 453–493. Kinokuniya, Tokyo, 1973.

  33. Hubbard J., Masur H.: Quadratic differentials and foliations. Acta Math., 142, 221–274 (1979)

    MATH  MathSciNet  Article  Google Scholar 

  34. Kapovich, M., Hyperbolic Manifolds and Discrete Groups. Progress in Mathematics, 183. Birkhäuser, Boston, MA, 2001.

  35. Kawai S.: The symplectic nature of the space of projective connections on Riemann surfaces. Math. Ann., 305, 161–182 (1996)

    MathSciNet  Article  Google Scholar 

  36. Kent R. P.: IV, Skinning maps. Duke Math. J., 151, 279–336 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  37. Kleinbock, D. & Weiss, B., Bounded geodesics in moduli space. Int. Math. Res. Not., 2004 (2004), 1551–1560.

  38. Kokorin, A. I. & Kopytov, V. M., Fully Ordered Groups. Halsted, New York–Toronto, 1974.

  39. Kra, I., Deformation spaces, in A Crash Course on Kleinian Groups (San Francisco, CA, 1974), Lecture Notes in Math., 400, pp. 48–70. Springer, Berlin–Heidelberg, 1974.

  40. Krasnov K., Schlenker J.-M.: A symplectic map between hyperbolic and complex Teichmüller theory. Duke Math. J., 150, 331–356 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  41. Labourie, F., Lectures on Representations of Surface Groups. Zürich Lectures in Advanced Mathematics. Eur. Math. Soc., Zürich, 2013.

  42. Levitt G.: Foliations and laminations on hyperbolic surfaces. Topology, 22, 119–135 (1983)

    MATH  MathSciNet  Article  Google Scholar 

  43. Levitt G.: \({\mathbb{R}}\)-trees and the Bieri–Neumann–Strebel invariant. Publ. Mat., 38, 195–202 (1994)

    MATH  MathSciNet  Article  Google Scholar 

  44. Masur H., Smillie J.: Hausdorff dimension of sets of nonergodic measured foliations. Ann. of Math., 134, 455–543 (1991)

    MATH  MathSciNet  Article  Google Scholar 

  45. McMullen C.: Amenability, Poincaré series and quasiconformal maps. Invent. Math., 97, 95–127 (1989)

    MATH  MathSciNet  Article  Google Scholar 

  46. McMullen C.: Iteration on Teichmüller space. Invent. Math., 99, 425–454 (1990)

    MATH  MathSciNet  Article  Google Scholar 

  47. Morgan, J. W., Group actions on trees and the compactification of the space of classes of SO(n, 1)-representations. Topology, 25 (1986), 1–33.

  48. Morgan J. W.: \({\Lambda}\)-trees and their applications. Bull. Amer. Math. Soc., 26, 87–112 (1992)

    MATH  MathSciNet  Article  Google Scholar 

  49. Morgan J. W., Shalen P. B.: Valuations, trees, and degenerations of hyperbolic structures. I. Ann. of Math., 120, 401–476 (1984)

    MATH  MathSciNet  Article  Google Scholar 

  50. Morgan, J. W. & Shalen, P. B., An introduction to compactifying spaces of hyperbolic structures by actions on trees, in Geometry and Topology (College Park, MD, 1983/84), Lecture Notes in Math., 1167, pp. 228–240. Springer, Berlin–Heidelberg, 1985.

  51. Morgan J. W., Shalen P. B.: Degenerations of hyperbolic structures. III. Actions of 3-manifold groups on trees and Thurston’s compactness theorem. Ann. of Math., 127, 457–519 (1988)

    MATH  MathSciNet  Article  Google Scholar 

  52. Morgan J. W., Shalen P. B.: Free actions of surface groups on R-trees.. Topology, 30, 143–154 (1991)

    MATH  MathSciNet  Article  Google Scholar 

  53. Mosher, L., Train track expansions of measured foliations. Preprint, 2003. http://andromeda.rutgers.edu/~mosher/arationality_03_12_28.pdf.

  54. Ohshika K.: Limits of geometrically tame Kleinian groups. Invent. Math., 99, 185–203 (1990)

    MATH  MathSciNet  Article  Google Scholar 

  55. Otal, J.-P., Thurston’s hyperbolization of Haken manifolds, in Surveys in Differential Geometry, Vol. III (Cambridge, MA, 1996), pp. 77–194. International Press, Boston, MA, 1998.

  56. Papadopoulos A.: Réseaux ferroviaires et courbes simples sur les surfaces. C. R. Acad. Sci. Paris Sér. I Math., 297, 565–568 (1983)

    MATH  Google Scholar 

  57. Papadopoulos A., Penner R. C.: The Weil–Petersson symplectic structure at Thurston’s boundary. Trans. Amer. Math. Soc., 335, 891–904 (1993)

    MATH  MathSciNet  Google Scholar 

  58. Penner, R. C. & Harer, J. L., Combinatorics of Train Tracks. Annals of Mathematics Studies, 125. Princeton Univ. Press, Princeton, NJ, 1992.

  59. Royden, H. L., Automorphisms and isometries of Teichmüller space, in Advances in the Theory of Riemann Surfaces (Stony Brook, NY, 1969), Ann. of Math. Studies, 66, pp. 369–383. Princeton Univ. Press, Princeton, NJ, 1971.

  60. Sullivan, D., On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, in Riemann Surfaces and Related Topics (Stony Brook, NY, 1978), Ann. of Math. Stud., 97, pp. 465–496. Princeton Univ. Press, Princeton, NJ, 1981.

  61. Veech W. A.: The Teichmüller geodesic flow. Ann. of Math., 124, 441–530 (1986)

    MATH  MathSciNet  Article  Google Scholar 

  62. Veech W. A.: Moduli spaces of quadratic differentials. J. Anal. Math., 55, 117–171 (1990)

    MATH  MathSciNet  Article  Google Scholar 

  63. Whitney H.: Tangents to an analytic variety. Ann. of Math., 81, 496–549 (1965)

    MATH  MathSciNet  Article  Google Scholar 

  64. Yue C. B.: Conditional measure and flip invariance of Bowen–Margulis and harmonic measures on manifolds of negative curvature. Ergodic Theory Dynam. Systems, 15, 807–811 (1995)

    MATH  MathSciNet  Article  Google Scholar 

  65. Zariski, O., & Samuel, P., Commutative Algebra. Vol. II. Graduate Texts in Mathematics, 29. Springer, New York–Heidelberg, 1975.

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Dumas, D. Skinning maps are finite-to-one. Acta Math 215, 55–126 (2015). https://doi.org/10.1007/s11511-015-0129-6

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Keywords

  • Length Function
  • Isometric Embedding
  • Compacti Cation
  • Saddle Connection
  • Measured Foliation