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

Maillot, Sylvain

doi:10.1090/S0002-9947-03-03319-1

Open 3-manifolds whose fundamental groups have infinite center, and a torus theorem for 3-orbifolds
HTML articles powered by AMS MathViewer

by Sylvain Maillot PDF

Trans. Amer. Math. Soc. 355 (2003), 4595-4638 Request permission

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.

References

Michel Boileau, Bernhard Leeb, and Joan Porti, On the geometry of $3$-dimensional cone manifolds, Preprint math.GT/0010185, 2000.
—, Uniformization of compact orientable $3$-orbifolds, Preprint math.GT/0010184, 2000.
Michel Boileau and Joan Porti, Geometrization of 3-orbifolds of cyclic type, Astérisque 272 (2001), 208 (English, with English and French summaries). Appendix A by Michael Heusener and Porti. MR 1844891
F. Bonahon and L. C. Siebenmann, The characteristic toric splitting of irreducible compact $3$-orbifolds, Math. Ann. 278 (1987), no. 1-4, 441–479. MR 909236, DOI 10.1007/BF01458079
Brian H. Bowditch, A topological characterisation of hyperbolic groups, J. Amer. Math. Soc. 11 (1998), no. 3, 643–667. MR 1602069, DOI 10.1090/S0894-0347-98-00264-1
—, Planar groups and the Seifert conjecture, Preprint, November 1999.
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
Daryl Cooper, Craig D. Hodgson, and Steven P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs, vol. 5, Mathematical Society of Japan, Tokyo, 2000. With a postface by Sadayoshi Kojima. MR 1778789
Warren Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge University Press, Cambridge, 1989. MR 1001965
William D. Dunbar, Hierarchies for $3$-orbifolds, Topology Appl. 29 (1988), no. 3, 267–283. MR 953958, DOI 10.1016/0166-8641(88)90025-9
M. J. Dunwoody and E. L. Swenson, The algebraic torus theorem, Invent. Math. 140 (2000), no. 3, 605–637. MR 1760752, DOI 10.1007/s002220000063
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
Wolfgang Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245–375 (German). MR 141106, DOI 10.1007/BF02559591
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, DOI 10.1090/cbms/043
William Jaco and J. Hyam Rubinstein, PL minimal surfaces in $3$-manifolds, J. Differential Geom. 27 (1988), no. 3, 493–524. MR 940116
William H. Jaco and Peter B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, DOI 10.1090/memo/0220
Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744, DOI 10.1007/BFb0085406
H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jber. Deutsch. Math.-Verein. 38 (1929), 248–260.
Sławomir Kwasik and Reinhard Schultz, Icosahedral group actions on $\textbf {R}^3$, Invent. Math. 108 (1992), no. 2, 385–402. MR 1161098, DOI 10.1007/BF02100611
Sylvain Maillot, Quasi-isometries of groups, graphs and surfaces, Comment. Math. Helv. 76 (2001), no. 1, 29–60. MR 1819660, DOI 10.1007/s000140050149
William H. Meeks III and Peter Scott, Finite group actions on $3$-manifolds, Invent. Math. 86 (1986), no. 2, 287–346. MR 856847, DOI 10.1007/BF01389073
John W. Morgan and Hyman Bass (eds.), The Smith conjecture, Pure and Applied Mathematics, vol. 112, Academic Press, Inc., Orlando, FL, 1984. Papers presented at the symposium held at Columbia University, New York, 1979. MR 758459
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
Geoffrey Mess, The Seifert conjecture and groups which are coarse quasiisometric to planes, Preprint.
G. P. Scott, Compact submanifolds of $3$-manifolds, J. London Math. Soc. (2) 7 (1973), 246–250. MR 326737, DOI 10.1112/jlms/s2-7.2.246
Peter Scott, Ends of pairs of groups, J. Pure Appl. Algebra 11 (1977/78), no. 1-3, 179–198. MR 487104, DOI 10.1016/0022-4049(77)90051-2
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 and Thomas Tucker, Some examples of exotic noncompact $3$-manifolds, Quart. J. Math. Oxford Ser. (2) 40 (1989), no. 160, 481–499. MR 1033220, DOI 10.1093/qmath/40.4.481
Yoshihiro Takeuchi, Partial solutions of the bad orbifold conjecture, Topology Appl. 72 (1996), no. 2, 113–120. MR 1404270, DOI 10.1016/0166-8641(96)00023-5
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 1696444, DOI 10.1016/S0166-8641(97)00280-0
Yoshihiro Takeuchi and Misako Yokoyama, PL-least area 2-orbifolds and its applications to 3-orbifolds, Kyushu J. Math. 55 (2001), no. 1, 19–61. MR 1825821, DOI 10.2206/kyushujm.55.19
William P. Thurston, The geometry and topology of three-manifolds, Lecture notes, Princeton University (1976–1979).
William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
—, Three-manifolds with symmetry, Preprint, 1982.
Pekka Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988), 1–54. MR 961162, DOI 10.1515/crll.1988.391.1
Friedhelm Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117 (German). MR 235576, DOI 10.1007/BF01402956
Friedhelm Waldhausen, Gruppen mit Zentrum und $3$-dimensionale Mannigfaltigkeiten, Topology 6 (1967), 505–517 (German). MR 236930, DOI 10.1016/0040-9383(67)90008-0
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594