Orderable 3-manifold groups
[Groupe de 3-variétés ordonnables]
Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 243-288.

On étudie l’ordonnabilité des groupes fondamentaux des variétés de dimension 3. Les groupes de nombreuses 3-variétés admettent un ordre invariant à gauche, y compris les groupes de toutes les variétés compactes P 2 -irréductibles dont le premier nombre de Betti est positif. Pour sept des huit géométries (toutes sauf l’hyperbolique) on caracterise exactement les variétés dont les groupes sont ordonnables à gauche, voire bi- ordonnables ; on démontre aussi qu’elles ont toutes des groupes virtuellement bi- ordonnables. L’ordonnabilité virtuelle en général, notamment pour les 3-variétés hyperboliques, est un problème qui reste ouvert et qui est lié à des conjectures de Waldhausen et d’autres auteurs.

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.

DOI : 10.5802/aif.2098
Classification : 57M05, 57M50, 20F60
Keywords: 3-manifold, orderable group, LO-group
Mot clés : 3 variétés, groupe ordonnable, groupe-LO
Boyer, Steven 1 ; Rolfsen, Dale  ; Wiest, Bert 

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)
@article{AIF_2005__55_1_243_0,
     author = {Boyer, Steven and Rolfsen, Dale and Wiest, Bert},
     title = {Orderable 3-manifold groups},
     journal = {Annales de l'Institut Fourier},
     pages = {243--288},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     doi = {10.5802/aif.2098},
     mrnumber = {2141698},
     zbl = {1068.57001},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2098/}
}
TY  - JOUR
AU  - Boyer, Steven
AU  - Rolfsen, Dale
AU  - Wiest, Bert
TI  - Orderable 3-manifold groups
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 243
EP  - 288
VL  - 55
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2098/
DO  - 10.5802/aif.2098
LA  - en
ID  - AIF_2005__55_1_243_0
ER  - 
%0 Journal Article
%A Boyer, Steven
%A Rolfsen, Dale
%A Wiest, Bert
%T Orderable 3-manifold groups
%J Annales de l'Institut Fourier
%D 2005
%P 243-288
%V 55
%N 1
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2098/
%R 10.5802/aif.2098
%G en
%F AIF_2005__55_1_243_0
Boyer, Steven; Rolfsen, Dale; Wiest, Bert. Orderable 3-manifold groups. Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 243-288. doi : 10.5802/aif.2098. http://archive.numdam.org/articles/10.5802/aif.2098/

[Ba] G. Baumslag On generalised free products, Math. Z., Volume 78 (1962), pp. 423-438 | DOI | MR | Zbl

[Be1] G. Bergman Left-orderable groups which are not locally indicable, Pac. J. Math., Volume 147 (1991), pp. 243-248 | MR | Zbl

[Be2] G. Bergman Ordering coproducts of groups and semigroups, J. Algebra, Volume 133 (1990), pp. 313-339 | DOI | MR | Zbl

[BH] R. Burns; V. Hale A note on group rings of certain torsion-free groups, Can. Math. Bull., Volume 15 (1972), pp. 441-445 | DOI | MR | Zbl

[Ca1] D. Calegari -covered foliations of hyperbolic 3-manifolds, Geometry and Topology, Volume 3 (1999), pp. 137-153 | DOI | MR | Zbl

[Ca2] D. Calegari The geometry of -covered foliations, Geom. Topol., Volume 4 (2000), pp. 457-515 | DOI | MR | Zbl

[CD] D. Calegari; N. Dunfield Laminations and groups of homeomorphisms of the circle, Invent. Math., Volume 152 (2003), pp. 149-207 | DOI | MR | Zbl

[CG] C. Champetier; V. Guirardel Limit groups as limits of free groups: compactifying the set of free groups (ArXiv math. GR/0401042, http://front.math.ucdavis.edu/math.GR/0401042)

[CJ] A. Casson; D. Jungreis Convergence groups and Seifert fibered 3-manifolds, Invent. Math., Volume 118 (1994), pp. 441-456 | DOI | MR | Zbl

[CK] I. Chiswell; P. Kropholler Soluble right orderable groups are locally indicable, Canad. Math. Bull., Volume 36 (1993), pp. 22-29 | DOI | MR | Zbl

[Co] P. F. Conrad Right-Ordered Groups, Michigan Math. J., Volume 6 (1959), pp. 267-275 | DOI | MR | Zbl

[De] P. Dehornoy Braid groups and left distributive operations, Trans. Amer. Math. Soc., Volume 345 (1994), pp. 115-150 | DOI | MR | Zbl

[DT] N. Dunfield; W. Thurston The virtual Haken conjecture: experiments and examples, Geom. Topol., Volume 7 (2003), pp. 399-441 | DOI | MR | Zbl

[Du] M. Dunwoody An equivariant sphere theorem, Bull. London Math. Soc., Volume 17 (1985), pp. 437-448 | DOI | MR | Zbl

[EHN] D. Eisenbud; U. Hirsch; W. Neumann Transverse foliations on Seifert bundles and self-homeomorphisms of the circle, Comm. Math. Helv., Volume 56 (1981), pp. 638-660 | DOI | MR | Zbl

[Ep] D. B. A. Epstein Projective planes in 3-manifolds, Proc. LMS, Volume 11 (1961), pp. 469-484 | MR | Zbl

[Fa] T. Farrell Right-orderable deck transformation groups, Rocky Mtn. J. Math., Volume 6 (1976), pp. 441-447 | DOI | MR | Zbl

[Fe] S. Fenley Anosov flows in 3-manifolds, Ann. Math., Volume 139 (1992) no. 2, pp. 79-115 | MR | Zbl

[FGRRW] R. Fenn; M. Greene; D. Rolfsen; C. Rourke; B. Wiest Ordering the braid groups, Pac. J. Math., Volume 191 (1999), pp. 49-74 | DOI | MR | Zbl

[G-M] J. González-Meneses Ordering pure braid groups on closed surfaces, Pac. J. Math., Volume 203 (2002), pp. 369-378 | DOI | MR | Zbl

[Ga1] D. Gabai Foliations and 3-manifolds (Proc. Int. Cong. Math.) (1990), pp. 609-619 | MR | Zbl

[Ga2] D. Gabai Convergence groups are Fuchsian groups, Ann. of Math., Volume 136 (1992), pp. 447-510 | DOI | MR | Zbl

[Ga3] D. Gabai Eight problems in the theory of foliations and laminations, Geometric Topology, 2, AMS/IP Studies in Advanced Mathematics, 1996 | MR

[GL] E. A. Gorin; V. Ya. Lin Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids, Math. USSR Sbornik, Volume 7 (1969), pp. 569-596 | DOI | MR | Zbl

[GS] A. M. Gaglione; D. Spellman Generalisations of free groups: some questions, Comm. Algebra, Volume 22 (1994), pp. 3159-3169 | DOI | MR | Zbl

[Ha] A. Hatcher Notes on basic 3-manifold topology (, http://www.math.cornell.edu/˜hatcher)

[He] J. Hempel 3-manifolds, Ann. of Math Studies, Volume 86 (1976) | MR | Zbl

[HiSt] P. Hilton; U. Stammbach A course in homological algebra, GTM, 4, 1971 | MR | Zbl

[HoSh] J. Howie; H. Short The band-sum problem, J. London Math. Soc., Volume 31 (1985), pp. 571-576 | DOI | MR | Zbl

[Jc] W. Jaco Lectures on three-manifold topology (CBMS Regional Conf. Ser. Math.), Volume 43 (1980) | MR | Zbl

[Jn] M. Jankins The space of homomorphisms of a Fuchsian groups to PSL 2 () (1983) (dissertation, University of Maryland) | MR

[JN1] M. Jankins; W. Neumann Homomorphisms of Fuchsian groups to PSL 2 (), Comm. Math. Helv., Volume 60 (1985), pp. 480-495 | DOI | MR | Zbl

[JN2] M. Jankins; W. Neumann Rotation numbers and products of circle homomorphisms, Math. Ann., Volume 271 (1985), pp. 381-400 | DOI | EuDML | MR | Zbl

[KR] D. Kim; D. Rolfsen An ordering for groups of pure braids and fibre-type hyperplane arrangements, Canad. J. Math., Volume 55 (2003), pp. 822-838 | DOI | MR | Zbl

[Li] P. Linnell Left ordered amenable and locally indicable groups, J. London Math. Soc., Volume 60 (1999) no. 2, pp. 133-142 | MR | Zbl

[Lo] D. Long Planar kernels in surface groups, Quart. J. Math. Oxford, Volume 35 (1984) no. 2, pp. 305-310 | MR | Zbl

[LR] R. H. La; Grange; A. H. Rhemtulla A remark on the group rings of order preserving permutation groups, Canad. Math. Bull, Volume 11 (1968), pp. 679-680 | DOI | MR | Zbl

[LS] R. Lyndon; P. Schupp Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 89, 1977 | MR | Zbl

[Lu] J. Luecke Finite covers of 3-manifolds containing essential tori, Trans. Amer. Math. Soc., Volume 310 (1988), pp. 381-391 | MR | Zbl

[Ma] A. I. Mal'cev On the embedding of group algebras in division algebras, Dokl. Akad. Nauk SSSR, Volume 60 (1948), p. 1944-1501 | MR | Zbl

[MKS] W. Magnus; A. Karrass; D. Solitar Combinatorial group theory, New York, 1976 | MR | Zbl

[MR] R. Botto; Mura; A. H. Rhemtulla Orderable groups, Lecture notes in pure and applied mathematics, 27, New York-Basel, 1977 | MR | Zbl

[MSY] W. Meeks; L. Simon; S. T. Yau Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math., Volume 116 (1982) no. 2, pp. 621-659 | MR | Zbl

[Na] R. Naimi Foliations transverse to fibers of Seifert manifolds, Comm. Math. Helv., Volume 69 (1994), pp. 155-162 | DOI | EuDML | MR | Zbl

[Ne] B. H. Neumann On ordered division rings, Trans. Amer. Math. Soc., Volume 66 (1949), pp. 202-252 | DOI | MR | Zbl

[No] S. P. Novikov Topology of foliations, Trans. Moscow Math. Soc., Volume 14 (1965), pp. 268-304 | MR | Zbl

[Pa] D. S. Passman The algebraic structure of group rings, Pure and applied mathematics, 1977 | MR | Zbl

[PR] B. Perron; D. Rolfsen On orderability of fibred knot groups (preprint) | MR | Zbl

[RoWi] D. Rolfsen; B. Wiest Free group automorphisms, invariant orderings and applications, Algebraic and Geometric Topology, Volume 1 (2001), pp. 311-319 | DOI | EuDML | MR | Zbl

[RR] A. Rhemtulla; D. Rolfsen Local indicability in ordered groups, braids and elementary amenable groups (preprint) | MR | Zbl

[RS] R. Roberts; M. Stein Exceptional Seifert group actions on , J. Knot Th. Ram., Volume 8 (1999), pp. 241-247 | DOI | MR | Zbl

[RSS] R. Roberts; J. Shareshian; M. Stein Infinitely many hyperbolic 3-manifolds which contain no Reebless Foliation to appear (JAMS) | MR | Zbl

[RW] C. Rourke; B. Wiest Order automatic mapping class groups, Pac. J. Math., Volume 194 (2000), pp. 209-227 | DOI | MR | Zbl

[RZ] D. Rolfsen; J. Zhu Braids, ordered groups and zero divisors, J. Knot Theory and Ramifications, Volume 7 (1998), pp. 837-841 | DOI | MR | Zbl

[Sc1] G. P. Scott Compact submanifolds of 3-manifolds, J. London Math. Soc., Volume 7 (1973), pp. 246-250 | DOI | MR | Zbl

[Sc2] G. P. Scott The geometries of 3-manifolds, Bull. Lond. Math. Soc., Volume 15 (1983), pp. 401-487 | DOI | MR | Zbl

[Sc3] G. P. Scott There are no fake Seifert fibre spaces with infinite π 1 , Ann. of Math., Volume 117 (1983), pp. 35-70 | DOI | MR | Zbl

[ScWa] G. P. Scott; T. Wall Topological methods in group theory, Homological group theory (Lecture Note Cambridge Univ. Press), Volume 36 (1977), pp. 137-203 | MR | Zbl

[Seif] H. Seifert Topologie dreidimensionaler gefaserter Räume, Acta Math., Volume 60 (1933), pp. 147-238 | DOI | JFM | MR | Zbl

[Sm] N. Smythe Trivial knots with arbitrary projection, J. Austral. Math. Soc., Volume 7 (1967), pp. 481-489 | DOI | MR | Zbl

[SW] H. Short; B. Wiest Orderings of mapping class groups after Thurston, Ens. Math., Volume 46 (2000), pp. 279-312 | MR | Zbl

[Th1] W. Thurston The geometry and topology of 3-manifolds, Lecture notes, 1977

[Th2] W. Thurston Three-manifolds, foliations and circles, I (1997) (preprint) | MR

[Tlf] J.\ Tollefson The compact 3-manifolds covered by S 2 ×, Proc. Amer. Math. Soc., Volume 45 (1974), pp. 461-462 | MR | Zbl

[V] A. A. Vinogradov On the free product of ordered groups, Mat. Sb., Volume 67 (1949), pp. 163-168 | MR | Zbl

Cité par Sources :