Geometriae Dedicata

, Volume 150, Issue 1, pp 181–232 | Cite as

Detection of incompressible surfaces in hyperbolic punctured torus bundles

Original Paper

Abstract

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher, and independently by Culler, Jaco and Rubinstein. We convert non fiber incompressible surfaces from their form to the form output by Yoshida’s construction, and run his construction backwards to give (for non semi-fibers, which we identify) the data needed to construct ideal points of the deformation variety corresponding to those surfaces via Yoshida’s construction. We use a result of Tillmann to show that the same incompressible surfaces can be obtained from an ideal point of the character variety via the Culler-Shalen construction. In particular this shows that all boundary slopes of non fiber and non semi-fiber incompressible surfaces in hyperbolic punctured torus bundles are strongly detected.

Keywords

Incompressible surfaces Punctured torus bundles Ideal point Triangulation Deformation variety 

Mathematics Subject Classification (2000)

57M99 

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References

  1. 1.
    Bergman G.M.: The logarithmic limit-set of an algebraic variety, Trans. Amer. Math. Soc. 157, 459–469 (1971)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chesebro E., Tillmann S.: Not all boundary slopes are strongly detected by the character variety. Comm. Anal. Geom. 15(4), 695–723. (2007) arXiv:math.GT/0510418MathSciNetMATHGoogle Scholar
  3. 3.
    Culler M., Jaco W., Rubinstein H.: Incompressible surfaces in once-punctured torus bundles. Proc. Lond. Math. Soc. 45(3), 385–419 (1982)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Culler M., Shalen P.: Varieties of group representations and splitting of 3-manifolds. Ann. Math. 117, 109–146 (1983)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Floyd W., Hatcher A.: Incompressible surfaces in punctured-torus bundles. Topol. Appl. 13, 263–282 (1982)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Guéritaud F., Futer D.: On canonical triangulations of once-punctured torus bundles and two-bridge link complements. Geom. Topol 10, 1239–1284 (2006) arXiv:math.GT/0406242MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Kabaya, Y.: A method to find ideal points from ideal triangulations. To appear in J. Knot Theory Ramifications. arXiv:math.GT/0706.0971.Google Scholar
  8. 8.
    Lackenby M.: The canonical decomposition of once-punctured torus bundles. Comment. Math. Helv. 78(2), 363–384 (2003)MathSciNetMATHGoogle Scholar
  9. 9.
    Mumford D.: The red book of varieties and schemes. 2nd edn. Springer, Berlin (1999)MATHGoogle Scholar
  10. 10.
    Ohtsuki T.: Ideal points and incompressible surfaces in two-bridge knot complements. J. Math. Soc. Japan 46(1), 51–87 (1994)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Schanuel S., Zhang X.: Detection of essential surfaces in 3-manifolds with SL2 trees. Mathematische Annalen 320(1), 149–165 (2001)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Segerman H.: On spun-normal and twisted squares surfaces. Proc. Am. Math. Soc. 137, 4259–4273 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Thurston, W.: The geometry and topology of 3-manifolds. Lecture Notes, Princeton University (1977)Google Scholar
  14. 14.
    Tillmann, S.: Degenerations of ideal hyperbolic triangulations. arXiv:math.GT/0508295Google Scholar
  15. 15.
    Yoshida T.: On ideal points of deformation curves of hyperbolic 3-manifolds with one cusp. Topology 30(2), 155–170 (1991)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.University of Texas at AustinAustinUSA

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