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Open 3-manifolds whose fundamental groups have infinite center, and a torus theorem for 3-orbifolds


Author: Sylvain Maillot
Journal: Trans. Amer. Math. Soc. 355 (2003), 4595-4638
MSC (2000): Primary 57N10
DOI: https://doi.org/10.1090/S0002-9947-03-03319-1
Published electronically: July 8, 2003
MathSciNet review: 1990764
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Abstract: Our main result is a characterization of open Seifert fibered $3$-manifolds in terms of the fundamental group and large-scale geometric properties of a triangulation. As an application, we extend the Seifert Fiber Space Theorem and the Torus Theorem to a class of $3$-orbifolds.


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  • 1. Michel Boileau, Bernhard Leeb, and Joan Porti, On the geometry of $3$-dimensional cone manifolds, Preprint math.GT/0010185, 2000.
  • 2. -, Uniformization of compact orientable $3$-orbifolds, Preprint math.GT/0010184, 2000.
  • 3. Michel Boileau and Joan Porti, Geometrization of 3-orbifolds of cyclic type, Astérisque (2001), no. 272, 208, Appendix A by Michael Heusener and Joan Porti. MR 2002f:57034
  • 4. F. Bonahon and L. C. Siebenmann, The characteristic toric splitting of irreducible compact $3$-orbifolds, Math. Ann. 278 (1987), 441-479. MR 90a:57017
  • 5. Brian H. Bowditch, A topological characterisation of hyperbolic groups, J. Amer. Math. Soc. 11 (1998), no. 3, 643-667. MR 99c:20048
  • 6. -, Planar groups and the Seifert conjecture, Preprint, November 1999.
  • 7. Andrew Casson and Douglas Jungreis, Convergence groups and Seifert fibered 3-manifolds, Invent. Math. 118 (1994), 441-456. MR 96f:57011
  • 8. Daryl Cooper, Craig D. Hodgson, and Steven P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds, Mathematical Society of Japan Memoirs, vol. 5, Japan Publ. Trading Co., 2000. MR 2002c:57027
  • 9. Warren Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge University Press, Cambridge, 1989. MR 91b:20001
  • 10. William D. Dunbar, Hierarchies for $3$-orbifolds, Topology Appl. 29 (1988), no. 3, 267-283. MR 89h:57008
  • 11. M. J. Dunwoody and E. L. Swenson, The algebraic torus theorem, Invent. Math. 140 (2000), no. 3, 605-637. MR 2001d:20039
  • 12. Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in 3-manifolds, Invent. Math. 71 (1983), 609-642. MR 85e:57012
  • 13. David Gabai, Convergence groups are Fuchsian groups, Annals of Math. 136 (1992), 447-510. MR 93m:20065
  • 14. Wolfgang Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245-375. MR 25:4519a
  • 15. John Hempel, 3-manifolds, Ann. of Math. Studies, vol. 86, Princeton University Press, Princeton, New Jersey, 1976. MR 54:3702
  • 16. William Jaco, Lectures on three-manifold topology, CBMS Lecture Notes, vol. 43, American Mathematical Society, Providence, RI, 1980. MR 81k:57009
  • 17. William Jaco and J. Hyam Rubinstein, PL minimal surfaces in 3-manifolds, J. Differential Geom. 27 (1988), no. 3, 493-524. MR 89e:57009
  • 18. William H. Jaco and Peter B. Shalen, Seifert fibered spaces in 3-manifolds, Memoirs of the American Mathematical Society, no. 220, American Mathematical Society, Providence, RI, September 1979. MR 81c:57010
  • 19. Klaus Johannson, Homotopy equivalences of 3-manifolds with boundary, Lecture Notes in Mathematics, vol. 761, Springer-Verlag, Berlin, 1979. MR 82c:57005
  • 20. H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jber. Deutsch. Math.-Verein. 38 (1929), 248-260.
  • 21. S\lawomir Kwasik and Reinhard Schultz, Icosahedral group actions on ${\mathbf {R}}^3$, Invent. Math. 108 (1992), no. 2, 385-402. MR 93b:57040
  • 22. Sylvain Maillot, Quasi-isometries of groups, graphs and surfaces, Comment. Math. Helv. 76 (2001), no. 1, 29-60. MR 2002e:20078
  • 23. William H. Meeks, III and Peter Scott, Finite group actions on $3$-manifolds, Invent. Math. 86 (1986), no. 2, 287-346. MR 88b:57039
  • 24. William H. Meeks, III and Shing-Tung Yau, Group actions on ${\mathbf{R}}^3$, The Smith conjecture (New York, 1979), Academic Press, Orlando, FL, 1984, pp. 167-179. MR 86i:57002
  • 25. William H. Meeks, III, Leon Simon, and Shing-Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Annals of Mathematics 116 (1982), 621-659. MR 84f:53053
  • 26. Geoffrey Mess, The Seifert conjecture and groups which are coarse quasiisometric to planes, Preprint.
  • 27. Peter Scott, Compact submanifolds of 3-manifolds, J. London Math. Soc. (2) 7 (1973), 246-250. MR 48:5080
  • 28. Peter Scott, Ends of pairs of groups, J. Pure Appl. Algebra 11 (1977/78), no. 1-3, 179-198. MR 81h:20047
  • 29. Peter Scott, There are no fake Seifert fibered spaces with infinite $\pi_1$, Annals of Math. 117 (1983), 35-70. MR 84c:57008
  • 30. Peter Scott and Thomas Tucker, Some examples of exotic noncompact $3$-manifolds, Quart. J. Math. Oxford Ser. (2) 40 (1989), no. 160, 481-499. MR 91b:57021
  • 31. Yoshihiro Takeuchi, Partial solutions of the bad orbifold conjecture, Topology Appl. 72 (1996), no. 2, 113-120. MR 98g:57020
  • 32. Yoshihiro Takeuchi and Misako Yokoyama, The geometric realizations of the decompositions of $3$-orbifold fundamental groups, Topology Appl. 95 (1999), no. 2, 129-153. MR 2000g:57028
  • 33. -, PL-least area 2-orbifolds and its applications to 3-orbifolds, Kyushu J. Math. 55 (2001), no. 1, 19-61. MR 2002k:57041
  • 34. William P. Thurston, The geometry and topology of three-manifolds, Lecture notes, Princeton University (1976-1979).
  • 35. -, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982), no. 3, 357-381. MR 83h:57019
  • 36. -, Three-manifolds with symmetry, Preprint, 1982.
  • 37. Pekka Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988), 1-54. MR 89m:30047
  • 38. Friedhelm Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten, Invent. Math. 3-4 (1967), 308-333, 87-117. MR 38:3880
  • 39. -, Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten, Topology 6 (1967), 505-517. MR 38:5223
  • 40. -, On irreducible 3-manifolds which are sufficiently large, Annals of Math. 87 (1968), 56-88. MR 36:7146

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Additional Information

Sylvain Maillot
Affiliation: Département de Mathématiques, Université du Québec à Montréal, Case postale 8888 succursale centre-ville, Montréal, Canada H3C 3P8
Email: maillot@math.uqam.ca

DOI: https://doi.org/10.1090/S0002-9947-03-03319-1
Received by editor(s): September 28, 2001
Received by editor(s) in revised form: November 25, 2002
Published electronically: July 8, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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