Geometry & Topology

Orderability and Dehn filling

Marc Culler and Nathan M Dunfield

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Motivated by conjectures relating group orderability, Floer homology and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on the fundamental groups of rational homology 3 –spheres. Specifically, for a compact 3 –manifold M with torus boundary, we give several criteria which imply that whole intervals of Dehn fillings of M have left-orderable fundamental groups. Our technique uses certain representations from π 1 ( M ) into PSL 2 ˜ , which we organize into an infinite graph in H 1 ( M ; ) called the translation extension locus. We include many plots of such loci which inform the proofs of our main results and suggest interesting avenues for future research.

Article information

Geom. Topol., Volume 22, Number 3 (2018), 1405-1457.

Received: 12 February 2016
Revised: 2 March 2017
Accepted: 15 April 2017
First available in Project Euclid: 31 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M60: Group actions in low dimensions
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M05: Fundamental group, presentations, free differential calculus 20F60: Ordered groups [See mainly 06F15]

orderable groups Dehn filling


Culler, Marc; Dunfield, Nathan M. Orderability and Dehn filling. Geom. Topol. 22 (2018), no. 3, 1405--1457. doi:10.2140/gt.2018.22.1405.

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  • S Basu, R Pollack, M-F Roy, Algorithms in real algebraic geometry, 2nd edition, Algorithms and Computation in Mathematics 10, Springer (2006)
  • A F Beardon, The geometry of discrete groups, Graduate Texts in Mathematics 91, Springer (1983)
  • M Bell, Flipper, version 0.9.8 (2015) Available at \setbox0\makeatletter\@url {\unhbox0
  • I Biswas, S Lawton, D Ramras, Fundamental groups of character varieties: surfaces and tori, Math. Z. 281 (2015) 415–425
  • J Bowden, Approximating $C^0$–foliations by contact structures, Geom. Funct. Anal. 26 (2016) 1255–1296
  • S Boyer, A Clay, Foliations, orders, representations, $L$–spaces and graph manifolds, Adv. Math. 310 (2017) 159–234
  • S Boyer, C M Gordon, L Watson, On $L$–spaces and left-orderable fundamental groups, Math. Ann. 356 (2013) 1213–1245
  • S Boyer, D Rolfsen, B Wiest, Orderable $3$–manifold groups, Ann. Inst. Fourier $($Grenoble$)$ 55 (2005) 243–288
  • S Boyer, X Zhang, On Culler–Shalen seminorms and Dehn filling, Ann. of Math. 148 (1998) 737–801
  • G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press, New York (1972)
  • B A Burton, The cusped hyperbolic census is complete, preprint (2014)
  • B A Burton, R Budney, W Pettersson, et al, Regina: software for $3$–manifold topology and normal surface theory (2014) Available at \setbox0\makeatletter\@url {\unhbox0
  • D Calegari, Real places and torus bundles, Geom. Dedicata 118 (2006) 209–227
  • D Calegari, scl, MSJ Memoirs 20, Mathematical Society of Japan, Tokyo (2009)
  • D Calegari, N M Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003) 149–204
  • D Calegari, A Walker, Ziggurats and rotation numbers, J. Mod. Dyn. 5 (2011) 711–746
  • P J Callahan, M V Hildebrand, J R Weeks, A census of cusped hyperbolic $3$–manifolds, Math. Comp. 68 (1999) 321–332
  • A Champanerkar, I Kofman, T Mullen, The $500$ simplest hyperbolic knots, J. Knot Theory Ramifications 23 (2014) art. id. 1450055
  • A Champanerkar, I Kofman, E Patterson, The next simplest hyperbolic knots, J. Knot Theory Ramifications 13 (2004) 965–987
  • F Charles, B Poonen, Bertini irreducibility theorems over finite fields, J. Amer. Math. Soc. 29 (2016) 81–94
  • A Clay, D Rolfsen, Ordered groups and topology, Graduate Studies in Mathematics 176, Amer. Math. Soc., Providence, RI (2016)
  • D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of $3$–manifolds, Invent. Math. 118 (1994) 47–84
  • D Cooper, D D Long, Remarks on the $A$–polynomial of a knot, J. Knot Theory Ramifications 5 (1996) 609–628
  • M Culler, N M Dunfield, M Goerner, J R Weeks, SnapPy, a computer program for studying the topology and geometry of $3$–manifolds Available at \setbox0\makeatletter\@url {\unhbox0
  • M Culler, P B Shalen, Varieties of group representations and splittings of $3$–manifolds, Ann. of Math. 117 (1983) 109–146
  • N M Dunfield, Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999) 623–657
  • N M Dunfield, The Mahler measure of the $A$–polynomial of $m129(0,3)$, preprint (2003) Appendix to D Boyd and F Rodriguez Villegas, Mahler's measure and the dilogarithm, II
  • N M Dunfield, Floer homology, orderable groups, and taut foliations of hyperbolic $3$–manifolds: an experimental study, workshop talk, IAS (December 2015) Available at \setbox0\makeatletter\@url {\unhbox0
  • N M Dunfield, D P Thurston, A random tunnel number one $3$–manifold does not fiber over the circle, Geom. Topol. 10 (2006) 2431–2499
  • D Eisenbud, U Hirsch, W Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv. 56 (1981) 638–660
  • D Gabai, Foliations and the topology of $3$–manifolds, III, J. Differential Geom. 26 (1987) 479–536
  • E Ghys, Groups acting on the circle, Enseign. Math. 47 (2001) 329–407
  • M Goerner, Trace field data for SnapPy manifolds, database (2015) Available at \setbox0\makeatletter\@url {\unhbox0
  • C M Gordon, Riley's conjecture on ${\rm SL}(2,\mathbb R)$ representations of $2$–bridge knots, J. Knot Theory Ramifications 26 (2017) art. id. 1740003
  • C Gordon, T Lidman, Taut foliations, left-orderability, and cyclic branched covers, Acta Math. Vietnam. 39 (2014) 599–635
  • C M Gordon, J Luecke, Reducible manifolds and Dehn surgery, Topology 35 (1996) 385–409
  • R Hakamata, M Teragaito, Left-orderable fundamental groups and Dehn surgery on genus one $2$–bridge knots, Algebr. Geom. Topol. 14 (2014) 2125–2148
  • J Hanselman, J Rasmussen, S D Rasmussen, L Watson, Taut foliations on graph manifolds, preprint (2015)
  • M Heusener, J Porti, The variety of characters in ${\rm PSL}_2(\mathbb C)$, Bol. Soc. Mat. Mexicana 10 (2004) 221–237
  • M Heusener, J Porti, Deformations of reducible representations of $3$–manifold groups into ${\rm PSL}_2(\mathbb C)$, Algebr. Geom. Topol. 5 (2005) 965–997
  • J Hoste, M Thistlethwaite, J Weeks, The first $1,701,936$ knots, Math. Intelligencer 20 (1998) 33–48
  • Y Hu, Left-orderability and cyclic branched coverings, Algebr. Geom. Topol. 15 (2015) 399–413
  • M Jankins, W D Neumann, Rotation numbers of products of circle homeomorphisms, Math. Ann. 271 (1985) 381–400
  • W H Kazez, R Roberts, $C^0$ approximations of foliations, preprint (2015)
  • V T Khoi, A cut-and-paste method for computing the Seifert volumes, Math. Ann. 326 (2003) 759–801
  • J Konvalina, V Matache, Palindrome-polynomials with roots on the unit circle, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004) 39–44
  • H Kraft, T Petrie, J D Randall, Quotient varieties, Adv. Math. 74 (1989) 145–162
  • T Lidman, L Watson, Nonfibered $L$–space knots, Pacific J. Math. 267 (2014) 423–429
  • C Maclachlan, A W Reid, The arithmetic of hyperbolic $3$–manifolds, Graduate Texts in Mathematics 219, Springer (2003)
  • K Mann, Rigidity and flexibility of group actions on the circle, preprint (2015)
  • R Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv. 69 (1994) 155–162
  • P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334
  • P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281–1300
  • J Porti, Torsion de Reidemeister pour les variétés hyperboliques, Mem. Amer. Math. Soc. 612, Amer. Math. Soc., Providence, RI (1997)
  • J Rasmussen, S D Rasmussen, Floer simple manifolds and $L$–space intervals, preprint (2015)
  • R Roberts, Taut foliations in punctured surface bundles, II, Proc. London Math. Soc. 83 (2001) 443–471
  • H Segerman, A generalisation of the deformation variety, Algebr. Geom. Topol. 12 (2012) 2179–2244
  • W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 339, Amer. Math. Soc., Providence, RI (1986) 99–130
  • W P Thurston, Three-manifolds, foliations and circles, I, preprint (1997)
  • S Tillmann, Degenerations of ideal hyperbolic triangulations, Math. Z. 272 (2012) 793–823
  • A T Tran, On left-orderability and cyclic branched coverings, J. Math. Soc. Japan 67 (2015) 1169–1178
  • A T Tran, On left-orderable fundamental groups and Dehn surgeries on knots, J. Math. Soc. Japan 67 (2015) 319–338
  • J Verschelde, Algorithm $795$: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Softw. 25 (1999) 251–276
  • J Verschelde, et al, PHCpack: solving polynomial systems via homotopy continuation (1999–2015) Available at \setbox0\makeatletter\@url {\unhbox0