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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia


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Volume 64, Issue 3

Issues

Low-dimensional topology and ordering groups

Dale Rolfsen
Published Online: 2014-07-05 | DOI: https://doi.org/10.2478/s12175-014-0228-4

Abstract

This expository paper explores the interaction of group ordering with topological questions, especially in dimensions 2 and 3. Among the topics considered are surfaces, braid groups, 3-manifolds and their structures such as foliations and mappings between them. A final section explores currently ongoing research regarding spaces of homeomorphisms and their orderability properties. This is not meant to be a comprehensive survey, but rather just a taste of the rich relationship between topology and the theory of ordered groups.

MSC: Primary 06F15, 20F60; Secondary 57M07, 57M27

Keywords: group ordering; manifold; surface; braid group; foliation

  • [1] AGOL, I.: The virtual haken conjecture. Preprint 2012, arXiv:1204.2810v1. Google Scholar

  • [2] ARTIN, E.: The theory of braids, Amer. Scientist 38 (1950), 112–119. Google Scholar

  • [3] BERGMAN, G. M.: Right orderable groups that are not locally indicable, Pacific J. Math. 147 (1991), 243–248. http://dx.doi.org/10.2140/pjm.1991.147.243CrossrefGoogle Scholar

  • [4] BOTTO MURA, R.— RHEMTULLA, A.: Orderable groups. Lect. Notes Pure Appl. Math. 27, Marcel Dekker Inc., New York, 1977. Google Scholar

  • [5] BOYER, S.— MCAGORDON, C.— WATSON, L.: On l-spaces and left-orderable fundamental groups. Preprint 2011, arXiv:1107.5016. Google Scholar

  • [6] BOYER, S.— ROLFSEN, D.— WIEST, B.: Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble) 55 (2005), 243–288. http://dx.doi.org/10.5802/aif.2098CrossrefGoogle Scholar

  • [7] BRODSKIĭ, S. D.: Equations over groups, and groups with one defining relation, Sibirsk. Mat. Zh. 25 (1984), 84–103. http://dx.doi.org/10.1007/BF00969512CrossrefGoogle Scholar

  • [8] BURNS, R. G.— HALE, V. W. D.: A note on group rings of certain torsion-free groups, Canad. Math. Bull. 15 (1972), 441–445. http://dx.doi.org/10.4153/CMB-1972-080-3CrossrefGoogle Scholar

  • [9] BUTTSWORTH, R. N.: A family of groups with a countable infinity of full orders, Bull. Aust. Math. Soc. 4 (1971), 97–104. http://dx.doi.org/10.1017/S000497270004630XCrossrefGoogle Scholar

  • [10] CALEGARI, D.: Foliations and the Geometry of 3-manifolds. Oxford Math. Monogr., Oxford University Press, Oxford, UK, 2007. Google Scholar

  • [11] CALEGARI, D.— DUNFIELD, N. M.: Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003), 149–204. http://dx.doi.org/10.1007/s00222-002-0271-6CrossrefGoogle Scholar

  • [12] CHEHATA, C. G.: An algebraically simple ordered group, Proc. Lond. Math. Soc. (3) 2 (1952), 183–197. http://dx.doi.org/10.1112/plms/s3-2.1.183CrossrefGoogle Scholar

  • [13] CLAY, A.— ROLFSEN, D.: Ordered groups, eigenvalues, knots, surgery and L-spaces, Math. Proc. Cambridge Philos. Soc. 152 (2012), 115–129. http://dx.doi.org/10.1017/S0305004111000557CrossrefGoogle Scholar

  • [14] CONRAD, P.: Right-ordered groups, Michigan Math. J. 6 (1959), 267–275. http://dx.doi.org/10.1307/mmj/1028998233CrossrefGoogle Scholar

  • [15] DEHORNOY, P.: Braid groups and left distributive operations, Trans. Amer. Math. Soc. 345 (1994), 115–150. http://dx.doi.org/10.1090/S0002-9947-1994-1214782-4CrossrefGoogle Scholar

  • [16] DEHORNOY, P.— DYNNIKOV, I.— ROLFSEN, D.— WIEST, B.: Ordering Braids, Math. Surveys and Monogr. 148. Amer. Math. Soc., Providence, RI, 2008. Google Scholar

  • [17] DUBROVINA, T. V.— DUBROVIN, N. I.: On braid groups, Mat. Sb. 192 (2001), 53–64. http://dx.doi.org/10.4213/sm564CrossrefGoogle Scholar

  • [18] FALK, M.— RANDELL, R.: Pure braid groups and products of free groups. In: Braids (Santa Cruz, CA, 1986). Contemp. Math. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 217–228. http://dx.doi.org/10.1090/conm/078/975081CrossrefGoogle Scholar

  • [19] FENN, R.— GREENE, M. T.— ROLFSEN, D.— ROURKE, C.— WIEST, B.: Ordering the braid groups, Pacific J. Math. 191 (1999), 49–74. http://dx.doi.org/10.2140/pjm.1999.191.49CrossrefGoogle Scholar

  • [20] FUCHS, L.: Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963. Google Scholar

  • [21] GABAI, D.— MEYERHOFF, R.— MILLEY, P.: Mom technology and volumes of hyperbolic 3-manifolds, Comment. Math. Helv. 86 (2011), 145–188. http://dx.doi.org/10.4171/CMH/221CrossrefWeb of ScienceGoogle Scholar

  • [22] GLASS, A. M. W.: Partially Ordered Groups. Series in Algebra, Vol. 7, World Scientific Publishing Co. Inc., River Edge, NJ, 1999. Google Scholar

  • [23] HÖLDER, O.: Die Axiome der Quantitat und die Lehre vom Mass, Ber. Verh. Sachs. Gesch. Wiss. Leipzig Math. Phys. Cl. 53 (1901), 1–64. Google Scholar

  • [24] HOWIE, J.— SHORT, H.: The band-sum problem, J. Lond. Math. Soc. (2) 31 (1985), 571–576. http://dx.doi.org/10.1112/jlms/s2-31.3.571CrossrefGoogle Scholar

  • [25] ITO, T.: Braid ordering and knot genus, J. Knot Theory Ramifications 20 (2011), 1311–1323. http://dx.doi.org/10.1142/S0218216511009169CrossrefGoogle Scholar

  • [26] JONES, V. F. R.: Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), 335–388. http://dx.doi.org/10.2307/1971403CrossrefGoogle Scholar

  • [27] KIM, D. M.— ROLFSEN, D.: An ordering for groups of pure braids and fibre-type hyperplane arrangements, Canad. J. Math. 55 (2003), 822–838. http://dx.doi.org/10.4153/CJM-2003-034-2CrossrefGoogle Scholar

  • [28] KOPYTOV, V. M.— MEDVEDEV, N. Ya.: Right-ordered Groups. Siberian School of Algebra and Logic, Consultants Bureau, New York, 1996. Google Scholar

  • [29] LEVI, F. W.: Contributions to the theory of ordered groups, Proc. Indian Acad. Sci. Sect. A. 17 (1943), 199–201. Google Scholar

  • [30] LINNELL, P. A.: The space of left orders of a group is either finite or uncountable, Bull. Lond. Math. Soc. 43 (2011), 200–202. http://dx.doi.org/10.1112/blms/bdq099CrossrefGoogle Scholar

  • [31] MALYUTIN, A. V.— NETSVETAEV, N. Yu.: Dehornoy order in the braid group and transformations of closed braids, Algebra i Analiz 15 (2003), No. 3, 170–187. Google Scholar

  • [32] MILNOR, J.: A unique decomposition theorem for 3-manifolds, Amer. J. Math. 84 (1962), 1–7. http://dx.doi.org/10.2307/2372800CrossrefGoogle Scholar

  • [33] NAVAS, A.: A finitely generated, locally indicable group with no faithful action by C 1diffeomorphisms of the interval, Geom. Topol. 14 (2010), 573–584. http://dx.doi.org/10.2140/gt.2010.14.573Web of ScienceCrossrefGoogle Scholar

  • [34] NAVAS, A.: On the dynamics of (left) orderable groups, Ann. Inst. Fourier (Grenoble) 60 (2010), 1685–1740. http://dx.doi.org/10.5802/aif.2570CrossrefGoogle Scholar

  • [35] NAVAS, A.— WIEST, B.: Nielsen-Thurston orders and the space of braid orderings, Bull. Lond. Math. Soc. 43 (2011), 901–911. http://dx.doi.org/10.1112/blms/bdr027CrossrefGoogle Scholar

  • [36] OZSVÁTH, P.— SZABÓ, Z.: On knot Floer homology and lens space surgeries, Topology 44 (2005), 1281–1300. http://dx.doi.org/10.1016/j.top.2005.05.001CrossrefGoogle Scholar

  • [37] PERRON, B.— ROLFSEN, D.: On orderability of fibred knot groups, Math. Proc. Cambridge Philos. Soc. 135 (2003), 147–153. http://dx.doi.org/10.1017/S0305004103006674CrossrefGoogle Scholar

  • [38] PRASOLOV, V. V.— SOSSINSKY, A. B.: Knots, Links, Braids and 3-manifolds. An Introduction to the new Invariants in Low-dimensional Topology. Transl. Math. Monogr. 154, Amer. Math. Soc., Providence, RI, 1997. Google Scholar

  • [39] RHEMTULLA, A.— ROLFSEN, D.: Local indicability in ordered groups: braids and elementary amenable groups, Proc. Amer. Math. Soc. 130 (2002), 2569–2577. http://dx.doi.org/10.1090/S0002-9939-02-06413-4CrossrefGoogle Scholar

  • [40] ROBERTS, R.— SHARESHIAN, J.— STEIN, M.: Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation, J. Amer. Math. Soc. 16 (2003), 639–679. http://dx.doi.org/10.1090/S0894-0347-03-00426-0Google Scholar

  • [41] ROLFSEN, D.: Knots and Links. Amer. Math. Soc. Chelsea Series 346, Amer. Math. Soc., Providence, RI, 2003. Google Scholar

  • [42] ROLFSEN, D.— WIEST, B.: Free group automorphisms, invariant orderings and topological applications, Algebr. Geom. Topol. 1 (2001), 311–320. http://dx.doi.org/10.2140/agt.2001.1.311CrossrefGoogle Scholar

  • [43] ROLFSEN, D.— ZHU, J.: Braids, orderings and zero divisors, J. Knot Theory Ramifications 7 (1998), 837–841. http://dx.doi.org/10.1142/S0218216598000425CrossrefGoogle Scholar

  • [44] SCOTT, G. P.: Compact submanifolds of 3-manifolds, J. Lond. Math. Soc. (2) 7 (1973), 246–250. http://dx.doi.org/10.1112/jlms/s2-7.2.246Google Scholar

  • [45] SHORT, H.— WIEST, B.: Orderings of mapping class groups after Thurston, Enseign. Math. (2) 46 (2000), 279–312. Google Scholar

  • [46] SIKORA, A. S.: Topology on the spaces of orderings of groups, Bull. Lond. Math. Soc. 36 (2004), 519–526. http://dx.doi.org/10.1112/S0024609303003060CrossrefGoogle Scholar

  • [47] THURSTON, W. P.: Three-dimensional Geometry and Topology, Vol. 1 (S. Levy, ed.), Princeton Math. Ser. 35 Princeton University Press, Princeton, NJ, 1997. Google Scholar

  • [48] VINOGRADOV, A. A.: On the free product of ordered groups, Mat. Sb. N.S. 25(67) (1949), 163–168. Google Scholar

  • [49] CALEGARI, D.— ROLFSEN, D.: Groups of PL homeomorphisms of cubes. Preprint, Jan. 2014, arXiv:1401.0570 Google Scholar

About the article

Published Online: 2014-07-05

Published in Print: 2014-06-01


Citation Information: Mathematica Slovaca, Volume 64, Issue 3, Pages 579–600, ISSN (Online) 1337-2211, DOI: https://doi.org/10.2478/s12175-014-0228-4.

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© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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