## Geometry & Topology

### Orderability and Dehn filling

#### Abstract

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 $PSL 2 ℝ ˜$, which we organize into an infinite graph in $H 1 ( ∂ M ; ℝ )$ 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

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

Dates
Revised: 2 March 2017
Accepted: 15 April 2017
First available in Project Euclid: 31 March 2018

https://projecteuclid.org/euclid.gt/1522461619

Digital Object Identifier
doi:10.2140/gt.2018.22.1405

Mathematical Reviews number (MathSciNet)
MR3780437

Zentralblatt MATH identifier
06864259

Keywords
orderable groups Dehn filling

#### Citation

Culler, Marc; Dunfield, Nathan M. Orderability and Dehn filling. Geom. Topol. 22 (2018), no. 3, 1405--1457. doi:10.2140/gt.2018.22.1405. https://projecteuclid.org/euclid.gt/1522461619

#### References

• 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 http://pypi.python.org/pypi/flipper {\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 http://tinyurl.com/regina3m {\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 http://snappy.computop.org {\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 https://youtu.be/VJi2T3jL1_Q {\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 http://ptolemy.unhyperbolic.org/ {\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 http://math.uic.edu/~jan {\unhbox0