AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Ordered Groups and Topology
About this Title
Adam Clay, University of Manitoba, Winnipeg, MB, Canada and Dale Rolfsen, University of British Columbia, Vancouver, BC, Canada
Publication: Graduate Studies in Mathematics
Publication Year:
2016; Volume 176
ISBNs: 978-1-4704-3106-8 (print); 978-1-4704-3562-2 (online)
DOI: https://doi.org/10.1090/gsm/176
MathSciNet review: MR3560661
MSC: Primary 57-02; Secondary 20F34, 20F36, 20F60, 57M07, 57M50
Table of Contents
Download chapters as PDF
Front/Back Matter
Chapters
- Chapter 1. Orderable groups and their algebraic properties
- Chapter 2. Hölder’s theorem, convex subgroups and dynamics
- Chapter 3. Free groups, surface groups and covering spaces
- Chapter 4. Knots
- Chapter 5. Three-dimensional manifolds
- Chapter 6. Foliations
- Chapter 7. Left-orderings of the braid groups
- Chapter 8. Groups of homeomorphisms
- Chapter 9. Conradian left-orderings and local indicability
- Chapter 10. Spaces of orderings
- Azer Akhmedov and Cody Martin, Non-bi-orderability of $6_{2}$ and $7_{6}$, 2015.
- J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275–306. MR 1501429, DOI 10.1090/S0002-9947-1928-1501429-1
- E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamberg 4 (1925), 47–72.
- E. Artin, Theory of braids, Ann. of Math. (2) 48 (1947), 101–126. MR 19087, DOI 10.2307/1969218
- Gilbert Baumslag, Some reflections on proving groups residually torsion-free nilpotent. I, Illinois J. Math. 54 (2010), no. 1, 315–325. MR 2776998
- George M. Bergman, Right orderable groups that are not locally indicable, Pacific J. Math. 147 (1991), no. 2, 243–248. MR 1084707
- Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
- Roberta Botto Mura and Akbar Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics, Vol. 27, Marcel Dekker, Inc., New York-Basel, 1977. MR 0491396
- Steven Boyer and Adam Clay, Foliations, orders, representations, L-spaces and graph manifolds, Preprint, available via http://arxiv.org/abs/1401.7726.
- Steven Boyer, Cameron McA. Gordon, and Liam Watson, On L-spaces and left-orderable fundamental groups, Math. Ann. 356 (2013), no. 4, 1213–1245. MR 3072799, DOI 10.1007/s00208-012-0852-7
- Steven Boyer, Dale Rolfsen, and Bert Wiest, Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 243–288 (English, with English and French summaries). MR 2141698
- S. D. Brodskiĭ, Equations over groups, and groups with one defining relation, Sibirsk. Mat. Zh. 25 (1984), no. 2, 84–103 (Russian). MR 741011
- R. G. Burns and V. W. D. Hale, A note on group rings of certain torsion-free groups, Canad. Math. Bull. 15 (1972), 441–445. MR 310046, DOI 10.4153/CMB-1972-080-3
- R. N. Buttsworth, A family of groups with a countable infinity of full orders, Bull. Austral. Math. Soc. 4 (1971), 97–104. MR 279013, DOI 10.1017/S000497270004630X
- Danny Calegari and Nathan M. Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003), no. 1, 149–204. MR 1965363, DOI 10.1007/s00222-002-0271-6
- Danny Calegari and Dale Rolfsen, Groups of PL homeomorphisms of cubes, 2014.
- Andrew J. Casson and Steven A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. MR 964685, DOI 10.1017/CBO9780511623912
- J. C. Cha and C. Livingston, Knotinfo: Table of knot invariants, http://www. indiana.edu/knotinfo.
- C. G. Chehata, An algebraically simple ordered group, Proc. London Math. Soc. (3) 2 (1952), 183–197. MR 47031, DOI 10.1112/plms/s3-2.1.183
- I. M. Chiswell, A. M. W. Glass, and John S. Wilson, Residual nilpotence and ordering in one-relator groups and knot groups, Math. Proc. Cambridge Philos. Soc. 158 (2015), no. 2, 275–288. MR 3310246, DOI 10.1017/S0305004114000644
- Adam Clay, Free lattice-ordered groups and the space of left orderings, Monatsh. Math. 167 (2012), no. 3-4, 417–430. MR 2961291, DOI 10.1007/s00605-011-0305-5
- Adam Clay, Colin Desmarais, and Patrick Naylor, Testing bi-orderability of knot groups, Preprint, available via http://arxiv.org/abs/1410.5774.
- Adam Clay and Dale Rolfsen, Ordered groups, eigenvalues, knots, surgery and $L$-spaces, Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 1, 115–129. MR 2860419, DOI 10.1017/S0305004111000557
- Paul Conrad, Right-ordered groups, Michigan Math. J. 6 (1959), 267–275. MR 106954
- Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Graduate Texts in Mathematics, No. 57, Springer-Verlag, New York-Heidelberg, 1977. Reprint of the 1963 original. MR 0445489
- Mieczysław K. Da̧bkowski, Józef H. Przytycki, and Amir A. Togha, Non-left-orderable 3-manifold groups, Canad. Math. Bull. 48 (2005), no. 1, 32–40. MR 2118761, DOI 10.4153/CMB-2005-003-6
- M. Dehn, Über die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910), no. 1, 137–168 (German). MR 1511580, DOI 10.1007/BF01455155
- Patrick Dehornoy, Monoids of $O$-type, subword reversing, and ordered groups, J. Group Theory 17 (2014), no. 3, 465–524. MR 3200370, DOI 10.1515/jgt-2013-0049
- Patrick Dehornoy, Ivan Dynnikov, Dale Rolfsen, and Bert Wiest, Ordering braids, Surveys and Monographs, vol. 148, American Mathematical Society, Providence, RI, 2008.
- B. Deroin, A. Navas, and C. Rivas, Groups, orders, and dynamics, 2014.
- T. V. Dubrovina and N. I. Dubrovin, On braid groups, Mat. Sb. 192 (2001), no. 5, 53–64 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 5-6, 693–703. MR 1859702, DOI 10.1070/SM2001v192n05ABEH000564
- David Eisenbud, Ulrich Hirsch, and Walter Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv. 56 (1981), no. 4, 638–660. MR 656217, DOI 10.1007/BF02566232
- D. B. A. Epstein, Periodic flows on three-manifolds, Ann. of Math. (2) 95 (1972), 66–82. MR 288785, DOI 10.2307/1970854
- Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
- F. Thomas Farrell, Right-orderable deck transformation groups, Rocky Mountain J. Math. 6 (1976), no. 3, 441–447. MR 418078, DOI 10.1216/RMJ-1976-6-3-441
- David Gabai, Robert Meyerhoff, and Peter Milley, Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22 (2009), no. 4, 1157–1215. MR 2525782, DOI 10.1090/S0894-0347-09-00639-0
- A. M. W. Glass, Partially ordered groups, Series in Algebra, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1791008, DOI 10.1142/3811
- E. A. Gorin and V. Ja. Lin, Algebraic equations with continuous coefficients, and certain questions of the algebraic theory of braids, Mat. Sb. (N.S.) 78 (120) (1969), 579–610 (Russian). MR 0251712
- Joshua Evan Greene, Alternating links and left-orderability, Preprint, available via http://arxiv.org/abs/1107.5232.
- Allen Hatcher, Notes on basic $3$-manifold topology, available from the author’s website, 2007, http://www.math.cornell.edu/ hatcher/3M/3M.pdf.
- G. Higman, The units of group rings, Proc. London Math. Soc. 46 (1940), no. 2, 231–248.
- Graham Higman, B. H. Neumann, and Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254. MR 32641, DOI 10.1112/jlms/s1-24.4.247
- Hugh M. Hilden, M. T. Lozano, and José María Montesinos, Universal knots, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 3, 449–450. MR 693959, DOI 10.1090/S0273-0979-1983-15114-5
- Hugh M. Hilden, María Teresa Lozano, and José María Montesinos, On knots that are universal, Topology 24 (1985), no. 4, 499–504. MR 816529, DOI 10.1016/0040-9383(85)90019-9
- Ya. V. Hion, Archimedean ordered rings, Uspehi Mat. Nauk 4 (1954), no. 9, 237–242.
- John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
- O. Hölder, Die Axiome der Quantität und die Lehre vom Mass, Ber. Verh. Sachs. Ges. Wiss. Leipzig Math. Phys. Cl. 53 (1901), 1–64.
- Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks, The first 1,701,936 knots, Math. Intelligencer 20 (1998), no. 4, 33–48. MR 1646740, DOI 10.1007/BF03025227
- James Howie, On locally indicable groups, Math. Z. 180 (1982), no. 4, 445–461. MR 667000, DOI 10.1007/BF01214717
- James Howie, A short proof of a theorem of Brodskiĭ, Publ. Mat. 44 (2000), no. 2, 641–647. MR 1800825, DOI 10.5565/publmat_{4}4200_{1}3
- James Howie and Hamish Short, The band-sum problem, J. London Math. Soc. (2) 31 (1985), no. 3, 571–576. MR 812788, DOI 10.1112/jlms/s2-31.3.571
- Edward V. Huntington, The continuum and other types of serial order. With an introduction to Cantor’s transfinite numbers, Dover Publications, Inc., New York, 1955. 2d ed. MR 0067953
- Tetsuya Ito, Construction of isolated left orderings via partially central cyclic amalgamation, Preprint, available via http://arxiv.org/abs/1107.0545.
- —, Isolated orderings on amalgamated free products, Preprint, available via http://arxiv.org/abs/1405.1163.
- Tetsuya Ito, Braid ordering and knot genus, J. Knot Theory Ramifications 20 (2011), no. 9, 1311–1323. MR 2844810, DOI 10.1142/S0218216511009169
- Tetsuya Ito, Dehornoy-like left orderings and isolated left orderings, J. Algebra 374 (2013), 42–58. MR 2998793, DOI 10.1016/j.jalgebra.2012.10.016
- Vu The Khoi, A cut-and-paste method for computing the Seifert volumes, Math. Ann. 326 (2003), no. 4, 759–801. MR 2003451, DOI 10.1007/s00208-003-0438-5
- Djun Maximilian Kim and Dale Rolfsen, An ordering for groups of pure braids and fibre-type hyperplane arrangements, Canad. J. Math. 55 (2003), no. 4, 822–838. MR 1994074, DOI 10.4153/CJM-2003-034-2
- Ali Ivanovič Kokorin and Valeriī Matveevič Kopytov, Fully ordered groups, Halsted Press [John Wiley & Sons], New York-Toronto; Israel Program for Scientific Translations, Jerusalem-London, 1974. Translated from the Russian by D. Louvish. MR 0364051
- V. M. Kopytov, Free lattice-ordered groups, Algebra i Logika 18 (1979), no. 4, 426–441, 508 (Russian). MR 582096
- Valeriĭ M. Kopytov and Nikolaĭ Ya. Medvedev, Right-ordered groups, Siberian School of Algebra and Logic, Consultants Bureau, New York, 1996. MR 1393199
- P. B. Kronheimer and T. S. Mrowka, Witten’s conjecture and property P, Geom. Topol. 8 (2004), 295–310. MR 2023280, DOI 10.2140/gt.2004.8.295
- F. W. Levi, Ordered groups, Proc. Indian Acad. Sci., Sect. A. 16 (1942), 256–263. MR 0007779
- F. W. Levi, Contributions to the theory of ordered groups, Proc. Indian Acad. Sci., Sect. A. 17 (1943), 199–201. MR 0008807
- W. B. R. Lickorish, A representation of orientable combinatorial $3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR 151948, DOI 10.2307/1970373
- W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963), 307–317. MR 145498, DOI 10.1017/s0305004100036926
- W. B. R. Lickorish, A foliation for $3$-manifolds, Ann. of Math. (2) 82 (1965), 414–420. MR 189061, DOI 10.2307/1970704
- Peter A. Linnell, Left ordered groups with no non-abelian free subgroups, J. Group Theory 4 (2001), no. 2, 153–168. MR 1812322, DOI 10.1515/jgth.2001.013
- Peter A. Linnell, The space of left orders of a group is either finite or uncountable, Bull. Lond. Math. Soc. 43 (2011), no. 1, 200–202. MR 2765563, DOI 10.1112/blms/bdq099
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, Second revised edition, Dover Publications, Inc., New York, 1976. Presentations of groups in terms of generators and relations. MR 0422434
- A. V. Malyutin and N. Yu. Netsvetaev, Dehornoy order in the braid group and transformations of closed braids, Algebra i Analiz 15 (2003), no. 3, 170–187 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 3, 437–448. MR 2052167, DOI 10.1090/S1061-0022-04-00816-7
- Stephen H. McCleary, Free lattice-ordered groups represented as $o$-$2$ transitive $l$-permutation groups, Trans. Amer. Math. Soc. 290 (1985), no. 1, 69–79. MR 787955, DOI 10.1090/S0002-9947-1985-0787955-7
- J. Milnor, A unique decomposition theorem for $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. MR 142125, DOI 10.2307/2372800
- Igor Mineyev, Submultiplicativity and the Hanna Neumann conjecture, Ann. of Math. (2) 175 (2012), no. 1, 393–414. MR 2874647, DOI 10.4007/annals.2012.175.1.11
- Dave Witte Morris, Amenable groups that act on the line, Algebr. Geom. Topol. 6 (2006), 2509–2518. MR 2286034, DOI 10.2140/agt.2006.6.2509
- James R. Munkres, Topology, 2nd edition, Prentice-Hall, 2000.
- Kunio Murasugi, Remarks on knots with two bridges, Proc. Japan Acad. 37 (1961), 294–297. MR 139162
- Andrés Navas, A finitely generated, locally indicable group with no faithful action by $C^1$ diffeomorphisms of the interval, Geom. Topol. 14 (2010), no. 1, 573–584. MR 2602845, DOI 10.2140/gt.2010.14.573
- Andrés Navas, On the dynamics of (left) orderable groups, Annales de l’institut Fourier 60 (2010), no. 5, 1685–1740.
- Andrés Navas, A remarkable family of left-ordered groups: central extensions of Hecke groups, J. Algebra 328 (2011), 31–42. MR 2745552, DOI 10.1016/j.jalgebra.2010.10.020
- B. H. Neumann, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202–252. MR 32593, DOI 10.1090/S0002-9947-1949-0032593-5
- B. H. Neumann, On ordered groups, Amer. J. Math. 71 (1949), 1–18. MR 28312, DOI 10.2307/2372087
- Yi Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007), no. 3, 577–608. MR 2357503, DOI 10.1007/s00222-007-0075-9
- S. P. Novikov, Foliations of co-dimension $1$ on manifolds, Dokl. Akad. Nauk SSSR 155 (1964), 1010–1013 (Russian). MR 0165540
- Peter Ozsváth and Zoltán Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005), no. 6, 1281–1300. MR 2168576, DOI 10.1016/j.top.2005.05.001
- Peter Ozsváth and Zoltán Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005), no. 1, 1–33. MR 2141852, DOI 10.1016/j.aim.2004.05.008
- C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1–26. MR 90053, DOI 10.2307/1970113
- Bernard Perron and Dale Rolfsen, On orderability of fibred knot groups, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 1, 147–153. MR 1990838, DOI 10.1017/S0305004103006674
- Akbar Rhemtulla and Dale Rolfsen, Local indicability in ordered groups: braids and elementary amenable groups, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2569–2577. MR 1900863, DOI 10.1090/S0002-9939-02-06413-4
- Cristóbal Rivas, Left-orderings on free products of groups, J. Algebra 350 (2012), 318–329. MR 2859890, DOI 10.1016/j.jalgebra.2011.10.036
- Cristóbal Rivas, On groups with finitely many Conradian orderings, Comm. Algebra 40 (2012), no. 7, 2596–2612. MR 2948849, DOI 10.1080/00927872.2011.583305
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR 1277811
- Dale Rolfsen and Bert Wiest, Free group automorphisms, invariant orderings and topological applications, Algebr. Geom. Topol. 1 (2001), 311–320. MR 1835259, DOI 10.2140/agt.2001.1.311
- Dale Rolfsen and Jun Zhu, Braids, orderings and zero divisors, J. Knot Theory Ramifications 7 (1998), no. 6, 837–841. MR 1643939, DOI 10.1142/S0218216598000425
- Colin Rourke and Bert Wiest, Order automatic mapping class groups, Pacific J. Math. 194 (2000), no. 1, 209–227. MR 1756636, DOI 10.2140/pjm.2000.194.209
- Rob Scharein, Knotplot, www.knotplot.com.
- Horst Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956), 133–170 (German). MR 82104, DOI 10.1007/BF01473875
- 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, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
- H. Seifert, Topologie Dreidimensionaler Gefaserter Räume, Acta Math. 60 (1933), no. 1, 147–238 (German). MR 1555366, DOI 10.1007/BF02398271
- Herbert Seifert and William Threlfall, Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, vol. 89, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. Translated from the German edition of 1934 by Michael A. Goldman; With a preface by Joan S. Birman; With “Topology of $3$-dimensional fibered spaces” by Seifert; Translated from the German by Wolfgang Heil. MR 575168
- Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
- Hamish Short and Bert Wiest, Orderings of mapping class groups after Thurston, Enseign. Math. (2) 46 (2000), no. 3-4, 279–312. MR 1805402
- Vladimir Shpilrain, Representing braids by automorphisms, Internat. J. Algebra Comput. 11 (2001), no. 6, 773–777. MR 1880377, DOI 10.1142/S0218196701000760
- Adam S. Sikora, Topology on the spaces of orderings of groups, Bull. London Math. Soc. 36 (2004), no. 4, 519–526. MR 2069015, DOI 10.1112/S0024609303003060
- N. Smythe, Trivial knots with arbitrary projection, J. Austral. Math. Soc. 7 (1967), 481–489. MR 0220271
- Masaaki Wada, Group invariants of links, Topology 31 (1992), no. 2, 399–406. MR 1167178, DOI 10.1016/0040-9383(92)90029-H
- Andrew H. Wallace, Modifications and cobounding manifolds, Canadian J. Math. 12 (1960), 503–528. MR 125588, DOI 10.4153/CJM-1960-045-7
- John W. Wood, Foliations on $3$-manifolds, Ann. of Math. (2) 89 (1969), 336–358. MR 248873, DOI 10.2307/1970673