Orderable 3-manifold groups

Steven Boyer[1]; Dale Rolfsen; Bert Wiest

  • [1] UQAM, Département de mathématiques, P.O. Box 8888, Centre-ville, Montréal, H3C 3P8, Québec (Canada), UBC, Department of Mathematics, Room 121, 1984 Mathematics Road, Vancouver V6T 1Z2 B.C. (Canada), Université de Rennes 1, Institut Mathématique, Campus de Beaulieu, 35042 Rennes Cedex (France)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 1, page 243-288
  • ISSN: 0373-0956

Abstract

top
We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P 2 -irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize which manifolds’ groups support a left-invariant or bi-invariant ordering. We also show that manifolds modelled on these geometries have virtually bi-orderable groups. The question of virtual orderability of 3-manifold groups in general, and even hyperbolic manifolds, remains open, and is closely related to conjectures of Waldhausen and others.

How to cite

top

Boyer, Steven, Rolfsen, Dale, and Wiest, Bert. "Orderable 3-manifold groups." Annales de l’institut Fourier 55.1 (2005): 243-288. <http://eudml.org/doc/116188>.

@article{Boyer2005,
abstract = {We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact $P^2$-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize which manifolds’ groups support a left-invariant or bi-invariant ordering. We also show that manifolds modelled on these geometries have virtually bi-orderable groups. The question of virtual orderability of 3-manifold groups in general, and even hyperbolic manifolds, remains open, and is closely related to conjectures of Waldhausen and others.},
affiliation = {UQAM, Département de mathématiques, P.O. Box 8888, Centre-ville, Montréal, H3C 3P8, Québec (Canada), UBC, Department of Mathematics, Room 121, 1984 Mathematics Road, Vancouver V6T 1Z2 B.C. (Canada), Université de Rennes 1, Institut Mathématique, Campus de Beaulieu, 35042 Rennes Cedex (France)},
author = {Boyer, Steven, Rolfsen, Dale, Wiest, Bert},
journal = {Annales de l’institut Fourier},
keywords = {3-manifold; orderable group; LO-group; left-orderable group},
language = {eng},
number = {1},
pages = {243-288},
publisher = {Association des Annales de l'Institut Fourier},
title = {Orderable 3-manifold groups},
url = {http://eudml.org/doc/116188},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Boyer, Steven
AU - Rolfsen, Dale
AU - Wiest, Bert
TI - Orderable 3-manifold groups
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 1
SP - 243
EP - 288
AB - We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact $P^2$-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize which manifolds’ groups support a left-invariant or bi-invariant ordering. We also show that manifolds modelled on these geometries have virtually bi-orderable groups. The question of virtual orderability of 3-manifold groups in general, and even hyperbolic manifolds, remains open, and is closely related to conjectures of Waldhausen and others.
LA - eng
KW - 3-manifold; orderable group; LO-group; left-orderable group
UR - http://eudml.org/doc/116188
ER -

References

top
  1. G. Baumslag, On generalised free products, Math. Z. 78 (1962), 423-438 Zbl0104.24402MR140562
  2. G. Bergman, Left-orderable groups which are not locally indicable, Pac. J. Math. 147 (1991), 243-248 Zbl0677.06007MR1084707
  3. G. Bergman, Ordering coproducts of groups and semigroups, J. Algebra 133 (1990), 313-339 Zbl0666.06012MR1067409
  4. R. Burns, V. Hale, A note on group rings of certain torsion-free groups, Can. Math. Bull. 15 (1972), 441-445 Zbl0244.16006MR310046
  5. D. Calegari, ℝ -covered foliations of hyperbolic 3-manifolds, Geometry and Topology 3 (1999), 137-153 Zbl0924.57014MR1695533
  6. D. Calegari, The geometry of ℝ -covered foliations, Geom. Topol. 4 (2000), 457-515 Zbl0964.57014MR1800151
  7. A. Casson, D. Jungreis, Convergence groups and Seifert fibered 3-manifolds, Invent. Math. 118 (1994), 441-456 Zbl0840.57005MR1296353
  8. D. Calegari, N. Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003), 149-207 Zbl1025.57018MR1965363
  9. C. Champetier, V. Guirardel, Limit groups as limits of free groups: compactifying the set of free groups Zbl1103.20026
  10. I. Chiswell, P. Kropholler, Soluble right orderable groups are locally indicable, Canad. Math. Bull. 36 (1993), 22-29 Zbl0802.20035MR1205890
  11. P. F. Conrad, Right-Ordered Groups, Michigan Math. J. 6 (1959), 267-275 Zbl0099.01703MR106954
  12. P. Dehornoy, Braid groups and left distributive operations, Trans. Amer. Math. Soc. 345 (1994), 115-150 Zbl0837.20048MR1214782
  13. N. Dunfield, W. Thurston, The virtual Haken conjecture: experiments and examples, Geom. Topol. 7 (2003), 399-441 Zbl1037.57015MR1988291
  14. M. Dunwoody, An equivariant sphere theorem, Bull. London Math. Soc. 17 (1985), 437-448 Zbl0592.57005MR806009
  15. D. Eisenbud, U. Hirsch, W. Neumann, Transverse foliations on Seifert bundles and self-homeomorphisms of the circle, Comm. Math. Helv. 56 (1981), 638-660 Zbl0516.57015MR656217
  16. D. B. A. Epstein, Projective planes in 3-manifolds, Proc. LMS 11 (1961), 469-484 Zbl0111.18801MR152997
  17. T. Farrell, Right-orderable deck transformation groups, Rocky Mtn. J. Math. 6 (1976), 441-447 Zbl0332.57004MR418078
  18. S. Fenley, Anosov flows in 3-manifolds, Ann. Math. 139 (1992), 79-115 Zbl0796.58039MR1259365
  19. R. Fenn, M. Greene, D. Rolfsen, C. Rourke, B. Wiest, Ordering the braid groups, Pac. J. Math. 191 (1999), 49-74 Zbl1009.20042MR1725462
  20. D. Gabai, Foliations and 3-manifolds, (1990), 609-619, Kyoto Zbl0754.57008
  21. D. Gabai, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992), 447-510 Zbl0785.57004MR1189862
  22. D. Gabai, Eight problems in the theory of foliations and laminations, 2 (1996), AMS/IP Studies in Advanced Mathematics 
  23. A. M. Gaglione, D. Spellman, Generalisations of free groups: some questions, Comm. Algebra 22 (1994), 3159-3169 Zbl0813.20023MR1272379
  24. J. González-Meneses, Ordering pure braid groups on closed surfaces, Pac. J. Math. 203 (2002), 369-378 Zbl1059.20033MR1897904
  25. E. A. Gorin, V. Ya. Lin, Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids, Math. USSR Sbornik 7 (1969), 569-596 Zbl0211.54905
  26. A. Hatcher, Notes on basic 3-manifold topology 
  27. J. Hempel, 3-manifolds, Ann. of Math Studies 86 (1976) Zbl0345.57001MR415619
  28. P. Hilton, U. Stammbach, A course in homological algebra, 4 (1971) Zbl0238.18006MR346025
  29. J. Howie, H. Short, The band-sum problem, J. London Math. Soc. 31 (1985), 571-576 Zbl0546.57001MR812788
  30. W. Jaco, Lectures on three-manifold topology, 43 (1980) Zbl0433.57001
  31. M. Jankins, The space of homomorphisms of a Fuchsian groups to P S L 2 ( ℝ ) , (1983) 
  32. M. Jankins, W. Neumann, Homomorphisms of Fuchsian groups to P S L 2 ( ℝ ) , Comm. Math. Helv. 60 (1985), 480-495 Zbl0598.57007MR814153
  33. M. Jankins, W. Neumann, Rotation numbers and products of circle homomorphisms, Math. Ann. 271 (1985), 381-400 Zbl0543.57019MR787188
  34. D. Kim, D. Rolfsen, An ordering for groups of pure braids and fibre-type hyperplane arrangements, Canad. J. Math. 55 (2003), 822-838 Zbl1047.20027MR1994074
  35. R. H. La, Grange, A. H. Rhemtulla, A remark on the group rings of order preserving permutation groups, Canad. Math. Bull 11 (1968), 679-680 Zbl0177.04502MR240183
  36. P. Linnell, Left ordered amenable and locally indicable groups, J. London Math. Soc. 60 (1999), 133-142 Zbl0940.20047MR1721820
  37. D. Long, Planar kernels in surface groups, Quart. J. Math. Oxford 35 (1984), 305-310 Zbl0556.57006MR755667
  38. J. Luecke, Finite covers of 3-manifolds containing essential tori, Trans. Amer. Math. Soc. 310 (1988), 381-391 Zbl0706.57009MR965759
  39. R. Lyndon, P. Schupp, Combinatorial group theory, 89 (1977) Zbl0368.20023MR577064
  40. W. Magnus, A. Karrass, D. Solitar, Combinatorial group theory, (1976), New York Zbl0362.20023MR422434
  41. A. I. Mal'cev, On the embedding of group algebras in division algebras, Dokl. Akad. Nauk SSSR 60 (1948), 1944-1501 Zbl0034.30901MR25457
  42. W. Meeks, L. Simon, S. T. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. 116 (1982), 621-659 Zbl0521.53007MR678484
  43. R. Botto, Mura, A. H. Rhemtulla, Orderable groups, 27 (1977), New York-Basel Zbl0358.06038
  44. R. Naimi, Foliations transverse to fibers of Seifert manifolds, Comm. Math. Helv. 69 (1994), 155-162 Zbl0797.55009MR1259611
  45. B. H. Neumann, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202-252 Zbl0035.30401MR32593
  46. S. P. Novikov, Topology of foliations, Trans. Moscow Math. Soc. 14 (1965), 268-304 Zbl0247.57006MR200938
  47. D. S. Passman, The algebraic structure of group rings, (1977) Zbl0368.16003MR470211
  48. B. Perron, D. Rolfsen, On orderability of fibred knot groups Zbl1046.57008MR1990838
  49. A. Rhemtulla, D. Rolfsen, Local indicability in ordered groups, braids and elementary amenable groups Zbl0996.20024MR1900863
  50. R. Roberts, J. Shareshian, M. Stein, Infinitely many hyperbolic 3-manifolds which contain no Reebless Foliation to appear Zbl1012.57022
  51. R. Roberts, M. Stein, Exceptional Seifert group actions on ℝ , J. Knot Th. Ram. 8 (1999), 241-247 Zbl0936.57014MR1687537
  52. D. Rolfsen, B. Wiest, Free group automorphisms, invariant orderings and applications, Algebraic and Geometric Topology 1 (2001), 311-319 Zbl0985.57006MR1835259
  53. D. Rolfsen, J. Zhu, Braids, ordered groups and zero divisors, J. Knot Theory and Ramifications 7 (1998), 837-841 Zbl0928.20031MR1643939
  54. C. Rourke, B. Wiest, Order automatic mapping class groups, Pac. J. Math. 194 (2000), 209-227 Zbl1016.57015MR1756636
  55. G. P. Scott, Compact submanifolds of 3-manifolds, J. London Math. Soc. 7 (1973), 246-250 Zbl0266.57001MR326737
  56. G. P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15 (1983), 401-487 Zbl0561.57001MR705527
  57. G. P. Scott, There are no fake Seifert fibre spaces with infinite π 1 , Ann. of Math. 117 (1983), 35-70 Zbl0516.57006MR683801
  58. G. P. Scott, T. Wall, Topological methods in group theory, Homological group theory, 36 (1977), 137-203, London Math. Soc., Cambridge-New York Zbl0423.20023
  59. H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1933), 147-238 Zbl0006.08304
  60. H. Short, B. Wiest, Orderings of mapping class groups after Thurston, Ens. Math. 46 (2000), 279-312 Zbl1023.57013MR1805402
  61. N. Smythe, Trivial knots with arbitrary projection, J. Austral. Math. Soc. 7 (1967), 481-489 Zbl0163.18102MR220271
  62. W. Thurston, The geometry and topology of 3-manifolds, (1977) 
  63. W. Thurston, Three-manifolds, foliations and circles, I, (1997) MR1435975
  64. J. Tollefson, The compact 3-manifolds covered by S 2 × ℝ , Proc. Amer. Math. Soc. 45 (1974), 461-462 Zbl0294.57006MR346792
  65. A. A. Vinogradov, On the free product of ordered groups, Mat. Sb. 67 (1949), 163-168 Zbl0038.15904MR31482

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.