Scott’s rigidity theorem for Seifert fibered spaces; revisited
HTML articles powered by AMS MathViewer
- by Teruhiko Soma PDF
- Trans. Amer. Math. Soc. 358 (2006), 4057-4070 Request permission
Abstract:
We will present a new proof of the rigidity theorem for Seifert fibered spaces of infinite $\pi _1$ by Scott (1983) in the case when the base of the fibration is a hyperbolic triangle 2-orbifold. Our proof is based on arguments in the rigidity theorem for hyperbolic 3-manifolds by Gabai (1997).References
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
- Michel Boileau and Jean-Pierre Otal, Groupe des difféotopies de certaines variétés de Seifert, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 1, 19–22 (French, with English summary). MR 849619
- Andrew Casson and Douglas Jungreis, Convergence groups and Seifert fibered $3$-manifolds, Invent. Math. 118 (1994), no. 3, 441–456. MR 1296353, DOI 10.1007/BF01231540
- Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), no. 3, 609–642. MR 695910, DOI 10.1007/BF02095997
- David Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510. MR 1189862, DOI 10.2307/2946597
- David Gabai, On the geometric and topological rigidity of hyperbolic $3$-manifolds, J. Amer. Math. Soc. 10 (1997), no. 1, 37–74. MR 1354958, DOI 10.1090/S0894-0347-97-00206-3
- David Gabai, G. Robert Meyerhoff, and Nathaniel Thurston, Homotopy hyperbolic 3-manifolds are hyperbolic, Ann. of Math. (2) 157 (2003), no. 2, 335–431. MR 1973051, DOI 10.4007/annals.2003.157.335
- John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
- William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
- William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. MR 678484, DOI 10.2307/2007026
- Peter Scott, There are no fake Seifert fibre spaces with infinite $\pi _{1}$, Ann. of Math. (2) 117 (1983), no. 1, 35–70. MR 683801, DOI 10.2307/2006970
- Peter Scott, Homotopy implies isotopy for some Seifert fibre spaces, Topology 24 (1985), no. 3, 341–351. MR 815484, DOI 10.1016/0040-9383(85)90006-0
- Teruhiko Soma, Existence of least area planes in hyperbolic 3-space with co-compact metric, Topology 43 (2004), no. 3, 705–716. MR 2041639, DOI 10.1016/j.top.2003.10.006
- Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
Additional Information
- Teruhiko Soma
- Affiliation: Department of Mathematical Sciences, College of Science and Engineering, Tokyo Denki University, Hatoyama-machi, Saitama-ken 350-0394, Japan
- MR Author ID: 192547
- Email: soma@r.dendai.ac.jp
- Received by editor(s): February 14, 2003
- Received by editor(s) in revised form: June 28, 2004
- Published electronically: September 22, 2005
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 4057-4070
- MSC (2000): Primary 57M99; Secondary 57M50
- DOI: https://doi.org/10.1090/S0002-9947-05-03804-3
- MathSciNet review: 2219010