Orderable 3-manifold groups

Boyer, Steven; Rolfsen, Dale; Wiest, Bert

Boyer, Steven; Rolfsen, Dale; Wiest, Bert

Annales de l'Institut Fourier, Volume 55 (2005) no. 1, p. 243-288

Abstract
Résumé

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.

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.

MR 2141698 | Zbl 1068.57001 | 1 citation in Numdam

DOI : https://doi.org/10.5802/aif.2098
Classification: 57M05, 57M50, 20F60
Keywords: 3-manifold, orderable group, LO-group

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     pages = {243-288},
     doi = {10.5802/aif.2098},
     zbl = {1068.57001},
     mrnumber = {2141698},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_1_243_0}
}

Boyer, Steven; Rolfsen, Dale; Wiest, Bert. Orderable 3-manifold groups. Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 243-288. doi : 10.5802/aif.2098. http://www.numdam.org/item/AIF_2005__55_1_243_0/

References

[Ba] G. Baumslag On generalised free products, Math. Z., Tome 78 (1962), pp. 423-438 | Article | MR 140562 | Zbl 0104.24402

[Be1] G. Bergman Left-orderable groups which are not locally indicable, Pac. J. Math., Tome 147 (1991), pp. 243-248 | MR 1084707 | Zbl 0677.06007

[Be2] G. Bergman Ordering coproducts of groups and semigroups, J. Algebra, Tome 133 (1990), pp. 313-339 | Article | MR 1067409 | Zbl 0666.06012

[BH] R. Burns; V. Hale A note on group rings of certain torsion-free groups, Can. Math. Bull., Tome 15 (1972), pp. 441-445 | Article | MR 310046 | Zbl 0244.16006

[Ca1] D. Calegari $ℝ$ -covered foliations of hyperbolic 3-manifolds, Geometry and Topology, Tome 3 (1999), pp. 137-153 | Article | MR 1695533 | Zbl 0924.57014

[Ca2] D. Calegari The geometry of $ℝ$ -covered foliations, Geom. Topol., Tome 4 (2000), pp. 457-515 | Article | MR 1800151 | Zbl 0964.57014

[CD] D. Calegari; N. Dunfield Laminations and groups of homeomorphisms of the circle, Invent. Math., Tome 152 (2003), pp. 149-207 | Article | MR 1965363 | Zbl 1025.57018

[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., Tome 118 (1994), pp. 441-456 | Article | MR 1296353 | Zbl 0840.57005

[CK] I. Chiswell; P. Kropholler Soluble right orderable groups are locally indicable, Canad. Math. Bull., Tome 36 (1993), pp. 22-29 | Article | MR 1205890 | Zbl 0802.20035

[Co] P. F. Conrad Right-Ordered Groups, Michigan Math. J., Tome 6 (1959), pp. 267-275 | Article | MR 106954 | Zbl 0099.01703

[De] P. Dehornoy Braid groups and left distributive operations, Trans. Amer. Math. Soc., Tome 345 (1994), pp. 115-150 | Article | MR 1214782 | Zbl 0837.20048

[DT] N. Dunfield; W. Thurston The virtual Haken conjecture: experiments and examples, Geom. Topol., Tome 7 (2003), pp. 399-441 | Article | MR 1988291 | Zbl 1037.57015

[Du] M. Dunwoody An equivariant sphere theorem, Bull. London Math. Soc., Tome 17 (1985), pp. 437-448 | Article | MR 806009 | Zbl 0592.57005

[EHN] D. Eisenbud; U. Hirsch; W. Neumann Transverse foliations on Seifert bundles and self-homeomorphisms of the circle, Comm. Math. Helv., Tome 56 (1981), pp. 638-660 | Article | MR 656217 | Zbl 0516.57015

[Ep] D. B. A. Epstein Projective planes in 3-manifolds, Proc. LMS, Tome 11 (1961), pp. 469-484 | MR 152997 | Zbl 0111.18801

[Fa] T. Farrell Right-orderable deck transformation groups, Rocky Mtn. J. Math., Tome 6 (1976), pp. 441-447 | Article | MR 418078 | Zbl 0332.57004

[Fe] S. Fenley Anosov flows in 3-manifolds, Ann. Math., Tome 139 (1992) no. 2, pp. 79-115 | MR 1259365 | Zbl 0796.58039

[FGRRW] R. Fenn; M. Greene; D. Rolfsen; C. Rourke; B. Wiest Ordering the braid groups, Pac. J. Math., Tome 191 (1999), pp. 49-74 | Article | MR 1725462 | Zbl 1009.20042

[G-M] J. González-Meneses Ordering pure braid groups on closed surfaces, Pac. J. Math., Tome 203 (2002), pp. 369-378 | Article | MR 1897904 | Zbl 1059.20033

[Ga1] D. Gabai Foliations and 3-manifolds, Kyoto (Proc. Int. Cong. Math.) (1990), pp. 609-619 | MR 1159248 | Zbl 0754.57008

[Ga2] D. Gabai Convergence groups are Fuchsian groups, Ann. of Math., Tome 136 (1992), pp. 447-510 | Article | MR 1189862 | Zbl 0785.57004

[Ga3] D. Gabai Eight problems in the theory of foliations and laminations, AMS/IP Studies in Advanced Mathematics, Geometric Topology, Tome 2 (1996) | MR 1470750

[GL] E. A. Gorin; V. Ya. Lin Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids, Math. USSR Sbornik, Tome 7 (1969), pp. 569-596 | Article | MR 251712 | Zbl 0211.54905

[GS] A. M. Gaglione; D. Spellman Generalisations of free groups: some questions, Comm. Algebra, Tome 22 (1994), pp. 3159-3169 | Article | MR 1272379 | Zbl 0813.20023

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

[He] J. Hempel 3-manifolds, Ann. of Math Studies, Tome 86 (1976) | MR 415619 | Zbl 0345.57001

[HiSt] P. Hilton; U. Stammbach A course in homological algebra, GTM, Tome 4 (1971) | MR 346025 | Zbl 0238.18006

[HoSh] J. Howie; H. Short The band-sum problem, J. London Math. Soc., Tome 31 (1985), pp. 571-576 | Article | MR 812788 | Zbl 0546.57001

[Jc] W. Jaco Lectures on three-manifold topology (CBMS Regional Conf. Ser. Math.) Tome 43 (1980) | MR 565450 | Zbl 0433.57001

[Jn] M. Jankins The space of homomorphisms of a Fuchsian groups to $P S L_{2} (ℝ)$ (1983) (dissertation, University of Maryland) | MR 2633143

[JN1] M. Jankins; W. Neumann Homomorphisms of Fuchsian groups to $P S L_{2} (ℝ)$ , Comm. Math. Helv., Tome 60 (1985), pp. 480-495 | Article | MR 814153 | Zbl 0598.57007

[JN2] M. Jankins; W. Neumann Rotation numbers and products of circle homomorphisms, Math. Ann., Tome 271 (1985), pp. 381-400 | Article | MR 787188 | Zbl 0543.57019 | Zbl 02234800

[KR] D. Kim; D. Rolfsen An ordering for groups of pure braids and fibre-type hyperplane arrangements, Canad. J. Math., Tome 55 (2003), pp. 822-838 | Article | MR 1994074 | Zbl 1047.20027

[Li] P. Linnell Left ordered amenable and locally indicable groups, J. London Math. Soc., Tome 60 (1999) no. 2, pp. 133-142 | MR 1721820 | Zbl 0940.20047

[Lo] D. Long Planar kernels in surface groups, Quart. J. Math. Oxford, Tome 35 (1984) no. 2, pp. 305-310 | MR 755667 | Zbl 0556.57006

[LR] R. H. La; Grange; A. H. Rhemtulla A remark on the group rings of order preserving permutation groups, Canad. Math. Bull, Tome 11 (1968), p. 679-680 | Article | MR 240183 | Zbl 0177.04502

[LS] R. Lyndon; P. Schupp Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Tome 89 (1977) | MR 577064 | Zbl 0368.20023

[Lu] J. Luecke Finite covers of 3-manifolds containing essential tori, Trans. Amer. Math. Soc., Tome 310 (1988), pp. 381-391 | MR 965759 | Zbl 0706.57009

[Ma] A. I. Mal'Cev On the embedding of group algebras in division algebras, Dokl. Akad. Nauk SSSR, Tome 60 (1948), p. 1944-1501 | MR 25457 | Zbl 0034.30901

[MKS] W. Magnus; A. Karrass; D. Solitar Combinatorial group theory, New York (1976) | MR 422434 | Zbl 0362.20023

[MR] R. Botto; Mura; A. H. Rhemtulla Orderable groups, New York-Basel, Lecture notes in pure and applied mathematics, Tome 27 (1977) | MR 491396 | Zbl 0358.06038

[MSY] W. Meeks; L. Simon; S. T. Yau Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math., Tome 116 (1982) no. 2, pp. 621-659 | MR 678484 | Zbl 0521.53007

[Na] R. Naimi Foliations transverse to fibers of Seifert manifolds, Comm. Math. Helv., Tome 69 (1994), pp. 155-162 | Article | MR 1259611 | Zbl 0797.55009

[Ne] B. H. Neumann On ordered division rings, Trans. Amer. Math. Soc., Tome 66 (1949), pp. 202-252 | Article | MR 32593 | Zbl 0035.30401

[No] S. P. Novikov Topology of foliations, Trans. Moscow Math. Soc., Tome 14 (1965), pp. 268-304 | MR 200938 | Zbl 0247.57006

[Pa] D. S. Passman The algebraic structure of group rings, Pure and applied mathematics (1977) | MR 470211 | Zbl 0368.16003

[PR] B. Perron; D. Rolfsen On orderability of fibred knot groups (preprint) | MR 1990838 | Zbl 1046.57008

[RoWi] D. Rolfsen; B. Wiest Free group automorphisms, invariant orderings and applications, Algebraic and Geometric Topology, Tome 1 (2001), pp. 311-319 | Article | MR 1835259 | Zbl 0985.57006

[RR] A. Rhemtulla; D. Rolfsen Local indicability in ordered groups, braids and elementary amenable groups (preprint) | MR 1900863 | Zbl 0996.20024

[RS] R. Roberts; M. Stein Exceptional Seifert group actions on $ℝ$ , J. Knot Th. Ram., Tome 8 (1999), pp. 241-247 | Article | MR 1687537 | Zbl 0936.57014

[RSS] R. Roberts; J. Shareshian; M. Stein Infinitely many hyperbolic 3-manifolds which contain no Reebless Foliation to appear (JAMS) | MR 1969207 | Zbl 1012.57022

[RW] C. Rourke; B. Wiest Order automatic mapping class groups, Pac. J. Math., Tome 194 (2000), pp. 209-227 | Article | MR 1756636 | Zbl 1016.57015

[RZ] D. Rolfsen; J. Zhu Braids, ordered groups and zero divisors, J. Knot Theory and Ramifications, Tome 7 (1998), pp. 837-841 | Article | MR 1643939 | Zbl 0928.20031

[Sc1] G. P. Scott Compact submanifolds of 3-manifolds, J. London Math. Soc., Tome 7 (1973), pp. 246-250 | Article | MR 326737 | Zbl 0266.57001

[Sc2] G. P. Scott The geometries of 3-manifolds, Bull. Lond. Math. Soc., Tome 15 (1983), pp. 401-487 | Article | MR 705527 | Zbl 0561.57001

[Sc3] G. P. Scott There are no fake Seifert fibre spaces with infinite $π_{1}$ , Ann. of Math., Tome 117 (1983), pp. 35-70 | Article | MR 683801 | Zbl 0516.57006

[ScWa] G. P. Scott; T. Wall Topological methods in group theory, Homological group theory, London Math. Soc., Cambridge-New York (Lecture Note Cambridge Univ. Press) Tome 36 (1977), pp. 137-203 | MR 564422 | Zbl 0423.20023

[Seif] H. Seifert Topologie dreidimensionaler gefaserter Räume, Acta Math., Tome 60 (1933), pp. 147-238 | Article | JFM 59.1241.02 | MR 1555366 | Zbl 0006.08304

[Sm] N. Smythe Trivial knots with arbitrary projection, J. Austral. Math. Soc., Tome 7 (1967), pp. 481-489 | Article | MR 220271 | Zbl 0163.18102

[SW] H. Short; B. Wiest Orderings of mapping class groups after Thurston, Ens. Math., Tome 46 (2000), pp. 279-312 | MR 1805402 | Zbl 1023.57013

[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 1435975

[Tlf] J.\ Tollefson The compact 3-manifolds covered by $S^{2} \times ℝ$ , Proc. Amer. Math. Soc., Tome 45 (1974), p. 461-462 | MR 346792 | Zbl 0294.57006

[V] A. A. Vinogradov On the free product of ordered groups, Mat. Sb., Tome 67 (1949), pp. 163-168 | MR 31482 | Zbl 0038.15904