Ordered groups, knots, braids and hyperbolic 3-manifolds

Rolfsen, Dale

Winter Braids VII (Caen, 2017), Winter Braids Lecture Notes (2017), Exposé no. 3, 24 p.

[1] Ian Agol. The minimal volume orientable hyperbolic 2-cusped 3-manifolds. Proc. Amer. Math. Soc., 138(10):3723–3732, 2010. | Zbl

[2] George M. Bergman. Right orderable groups that are not locally indicable. Pacific J. Math., 147(2):243–248, 1991. | Zbl

[3] Danny Calegari and Nathan M. Dunfield. Laminations and groups of homeomorphisms of the circle. Invent. Math., 152(1):149–204, 2003. | Zbl

[4] Chun Cao and G. Robert Meyerhoff. The orientable cusped hyperbolic $3$-manifolds of minimum volume. Invent. Math., 146(3):451–478, 2001. | DOI | Zbl

[5] Adam Clay, Colin Desmarais, and Patrick Naylor. Testing bi-orderability of knot groups. Canad. Math. Bull., 59(3):472–482, 2016. | Zbl

[6] Adam Clay and Dale Rolfsen. Ordered groups, eigenvalues, knots, surgery and $L$-spaces. Math. Proc. Cambridge Philos. Soc., 152(1):115–129, 2012. | Zbl

[7] M. Dehn. Über die Topologie des dreidimensionalen Raumes. Math. Ann., 69(1):137–168, 1910. | DOI | Zbl

[8] Patrick Dehornoy. Braid groups and left distributive operations. Trans. Amer. Math. Soc., 345(1):115–150, 1994. | Zbl

[9] Bertrand Deroin, Andrés Navas, and Cristóbal Rivas. Groups, orders and dynamics. 2014. Available at arXiv:1408.5805v1.

[10] David Gabai, Robert Meyerhoff, and Peter Milley. Minimum volume cusped hyperbolic three-manifolds. J. Amer. Math. Soc., 22(4):1157–1215, 2009. | Zbl

[11] Juan González-Meneses. Basic results on braid groups. Ann. Math. Blaise Pascal, 18(1):15–59, 2011.

[12] John Hempel. $3$-Manifolds. Princeton University Press, Princeton, N. J., 1976. Ann. of Math. Studies, No. 86. | Zbl

[13] O. Hölder. Die Axiome der Quantität und die Lehre vom Mass. Ber. Verh. Sachs. Ges. Wiss. Leipzig Math. Phys. Cl., 53:1–64, 1901.

[14] James Howie and Hamish Short. The band-sum problem. J. London Math. Soc. (2), 31(3):571–576, 1985. | Zbl

[15] Tetsuya Ito. Alexander polynomial obstruction of bi-orderability for rationally homologically fibered knot groups. New York J. Math., 23:497–503, 2017. | Zbl

[16] Eiko Kin and Dale Rolfsen. Braids, orderings and minimal volume cusped hyperbolic 3-manifolds. Groups, Geometry and Dynamics. Electronically published on July 27, 2018. doi: 10.4171/GGD/460 (to appear in print). | DOI | Zbl

[17] R.H. La Grange and A.H. Rhemtulla. A remark on the group rings of order preserving permutation groups. Canad. Math. Bull., 11:679–680, 1968. | DOI | Zbl

[18] F. W. Levi. Contributions to the theory of ordered groups. Proc. Indian Acad. Sci., Sect. A., 17:199–201, 1943. | Zbl

[19] Yi Ni. Knot Floer homology detects fibred knots. Invent. Math., 170(3):577–608, 2007. | Zbl

[20] Peter Ozsváth and Zoltán Szabó. On knot Floer homology and lens space surgeries. Topology, 44(6):1281–1300, 2005.

[21] Bernard Perron and Dale Rolfsen. On orderability of fibred knot groups. Math. Proc. Cambridge Philos. Soc., 135(1):147–153, 2003. | Zbl

[22] Dale Rolfsen. Ordered groups as a tensor category. Pacific J. Math., 294(1):181–194, 2018. | Zbl

[23] Dale Rolfsen and Bert Wiest. Free group automorphisms, invariant orderings and topological applications. Algebr. Geom. Topol., 1:311–320 (electronic), 2001. | Zbl

[24] Dale Rolfsen and Jun Zhu. Braids, orderings and zero divisors. J. Knot Theory Ramifications, 7(6):837–841, 1998. | DOI | Zbl

[25] G. P. Scott. Compact submanifolds of $3$-manifolds. J. London Math. Soc. (2), 7:246–250, 1973. | Zbl

[26] A.S. Sikora. Topology on the spaces of orderings of groups. Bull. London Math. Soc., 36:519–526, 2004. | Zbl

[27] A. A. Vinogradov. On the free product of ordered groups. Mat. Sbornik N.S., 25(67):163–168, 1949.

[28] Ken’ichi Yoshida. The minimal volume orientable hyperbolic 3-manifold with 4 cusps. Pacific J. Math., 266(2):457–476, 2013. | Zbl